Sound Waves and Music

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he Physics Classroom

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Sound Wavesand Music

Lesson 1

Lesson 2

Lesson 3

Lesson 4

Lesson 5

Sound Waves and Music

The nature of sound as a longitudinal,mechanical pressure wave is explained andthe properties of sound are discussed.Wave principles of resonance and standingwaves are applied in an effort to analyzethe physics of musical instruments.

Lesson 1: The Nature of a Sound Wave

Lesson 2: Sound Properties and TheirPerception

Lesson 3: Behavior of Sound Waves

Lesson 4: Resonance and StandingWaves

Lesson 5: Musical Instruments

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he Physics Classroom Page 1

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Lesson 1

Sound is aMechanicalWave

Sound is aLongitudinalWave

Sound is aPressureWave

Lesson 2

Lesson 3

Lesson 4

Lesson 5


Sound is a Mechanical Wave

Sound and music are parts of our everydaysensory experience. Just as humans haveeyes for the detection of light and color, sowe are equipped with ears for the detectionof sound.We seldom take the time to

ponder the characteristics and behaviors of sound and the mechanisms by whichsounds are produced, propagated, anddetected. The basis for an understanding of sound, music and hearing is the physics of waves. Sound is a wave which is created byvibrating objects and propagated through amedium from one location to another. Inthis unit, we will investigate the nature,properties and behaviors of sound waves

and apply basic wave principles towards anunderstanding of music.

As discussed in the previous unit of ThePhysics Classroom, a wave can be describedas a disturbance that travels through amedium, transporting energy from onelocation to another location. The medium issimply the material through which thedisturbance is moving; it can be thought of

as a series of interacting particles. Theexample of a slinky wave is often used toillustrate the nature of a wave. Adisturbance is typically created within theslinky by the back and forth movement of the first coil of the slinky. The first coilbecomes disturbed and begins to push orpull on the second coil; this push or pull onthe second coil will displace the second coilfrom its equilibrium position. As the second

coil becomes displaced, it begins to push orpull on the third coil; the push or pull onthe third coil displaces it from itsequilibrium position. As the third coil




becomes displaced, it begins to push or pullon the fourth coil. This process continues inconsecutive fashion, each individual particleacting to displace the adjacent particle;

subsequently the disturbance travelsthrough the slinky. As the disturbancemoves from coil to coil, the energy whichwas originally introduced into the first coilis transported along the medium from onelocation to another.

A sound wave is similar in nature to a slinkywave for a variety of reasons. First, there isa medium which carries the disturbancefrom one location to another. Typically, thismedium is air; though it could be anymaterial such as water or steel. Themedium is simply a series of interconnectedand interacting particles. Second, there isan original source of the wave, somevibrating object capable of disturbing thefirst particle of the medium. The vibratingobject which creates the disturbance couldbe the vocal chords of a person, thevibrating string and sound board of a guitaror violin, the vibrating tines of a tuningfork, or the vibrating diaphragm of a radiospeaker. Third, the sound wave istransported from one location to another bymeans of the particle interaction. If thesound wave is moving through air, then asone air particle is displaced from its

equilibrium position, it exerts a push or pullon its nearest neighbors, causing them tobe displaced from their equilibrium position.This particle interaction continuesthroughout the entire medium, with eachparticle interacting and causing adisturbance of its nearest neighbors. Sincea sound wave is a disturbance which istransported through a medium via themechanism of particle interaction, a sound

wave is characterized as a mechanicalwave.




The creation andpropagation of soundwaves are often

demonstrated in classthrough the use of a tuningfork. A tuning fork is ametal object consisting of two tines capable of vibrating if struck by arubber hammer or mallet. As the tines of the tuning forks vibrate back and forth,they begin to disturb surrounding airmolecules. These disturbances are passedon to adjacent air molecules by the

mechanism of particle interaction. Themotion of the disturbance, originating atthe tines of the tuning fork and travelingthrough the medium (in this case, air) iswhat is referred to as a sound wave. Thegeneration and propagation of a soundwave is demonstrated in the animationbelow.

In some class demonstrations, the tuningfork is mounted on a sound board. In suchinstances, the vibrating tuning fork, beingconnected to the sound board, sets thesound board into vibrational motion. Inturn, the sound board, being connected tothe air inside of it, sets the air inside of thesound board into vibrational motion. As thetines of the tunig fork, the structure of thesound board, and the inside of the soundboard begin vibrating at the samefrequency, a louder sound is produced. Infact, the more particles which can be madeto vibrate, the louder or more amplified thesound. This concept was also demonstratedby the placement of the vibrating tunig forkagainst the glass panel of the overheadprojector; the vibrating tuning fork set theglass panel into vibrational motion andresulted in an amplified sound.




In the tuning fork demonstrations, we knowthat the tuning fork is vibrating because wehear the sound which is produced by their

vibration. Nonetheless, we do not actuallyvisibly detect any vibrations of the tines.This is because the tines arevibrating at a very highfrequency. If the tuning forkwhich is being usedcorresponds to middle C onthe piano keyboard, then the tines arevibrating at a frequency of 256 Hz - 256vibrations per second. We are unable to

detect vibrations of such high frequency.But perhaps you recall the demonstration inwhich a high frequency strobe light wasused to slow down the vibrations. If hestrobe light puts out a flash of light at afrequency of 512 Hz (two times thefrequency of the tuning fork), then thetuning fork can be observed to be moving ina back and forth motion. With the roomdarkened, the strobe allows us to view the

position of the tines two times during theirvibrational cycle. Thus we see the tineswhen they are displaced far to the left andagain when they are displaced far to theright. This is convincing proof that the tinesof the tuning fork are indeed vibrating.

In a previous unit of The Physics Classroom,a distinction was made between twocategories of waves: mechanical waves and

electromagnetic waves. Electromagneticwaves are waves which have an electric andmagnetic nature and arecapable of travelingthrough a vacuum.Electromagnetic wavesdo not require a mediumin order to transport theirenergy. Mechanicalwaves are waves which require a medium

in order to transport their energy from onelocation to another. Because mechanicalwaves rely on particle interaction in orderto transport their energy, they cannot




travel through regions of space which aredevoid of particles. That is, mechanicalwaves cannot travel through a vacuum.This feature of mechanical waves was

demonstrated in class using a segmentfrom a laser disc. A ringing bell was plaedin a jar and air was evacuated from the jar.Once air was removed from the jar, thesound of the ringing bell could no longer beheard. The clapper could be seen strikingthe bell. but the sound which it producedcould not be heard because there were noparticles inside of the jar to transport thedisturbance through the vacuum. Sound is

a mechanical wave and cannot travelthrough a vacuum.

Check Your Undersanding

1. A sound wave is different than a lightwave in that a sound wave is

a. produced by an oscillating object and a

light wave is not.b. not capable of traveling through a

vacuum.

c. not capable of diffracting and a lightwave is.

d. capable of existing with a variety of frequencies and a light wave has asingle frequency.

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Lesson 1




Lesson 2

Lesson 3

Lesson 4

Lesson 5


Sound is a Longitudinal Wave

In the first part of Lesson 1, it wasmentioned that sound is a mechanical wavewhich is created by a vibrating object. Thevibrations of the object set particles in thesurrounding medium in vibrational motion,

thus transporting energy through themedium. The vibrations of the particles arebest described as longitudinal.Longitudinal waves are waves in which themotion of the individual particles of themedium is in a direction which is parallel tothe direction of energy transport. Alongitudinal wave can be created in a slinkyif the slinky is stretched out in a horizontaldirection and the first coils of the slinky are

vibrated horizontally. In such a case, eachindividual coil of the medium is set intovibrational motion in directions parallel tothe direction which the energy istransported.

Sound waves are longitudinal wavesbecause particles of the medium throughwhich the sound is transported vibrateparallel to the direction which the soundmoves. A vibrating string can createlongitudinal waves as depicted in theanimation below. As the vibrating string

moves in the forward direction, it begins topush upon surrounding air molecules,moving them to the right towards theirnearest neighbor. This causes the air




molecules to the right of the string to becompressed into a small region of space. Asthe vibrating string moves in the reversedirection (leftward), it lowers the pressure

of the air immediately to its right, thuscausing air molecules to move backleftward. The lower pressure to the right of the string causes air molecules in thatregion immediately to the right of the stringto expand into a large region of space. Theback and forth vibration of the string causesindividual air molecules (or a layer of airmolecules) in the region immediately to theright of the string to continually move back

and forth horizontally; the molecules moverightward as the string moves rightwardand then leftward as the string movesleftward. These back and forth vibrationsare imparted to adjacent neighbors byparticle interaction; thus, other surroundingparticles begin to move rightward andleftward, thus sending a wave to the right.Since air molecules (the particles of themedium) are moving in a direction which is

parallel to the direction which the wavemoves, the sound wave is referred to as alongitudinal wave. The result of suchlongitudinal vibrations is the creation of compressions and rarefactions withinthe air.

Regardless of the source of the sound wave- whether it be a vibrating string or thevibrating tines of a tuning fork - sound is alongitudinal wave. And the essentialcharacteristic of a longitudinal wave whichdistinguishes it from other types of waves is

that the particles of the medium move in adirection parallel to the direction of energytransport.




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Lesson 1




Lesson 2

Lesson 3

Lesson 4

Lesson 5


Sound is a Pressure Wave

Sound is a mechanical wave which resultsfrom the longitudinal motion of the particlesof the medium through which the soundwave is moving. If a sound wave is movingfrom left to right through air, then particles

of air will be displaced both rightward andleftward as the energy of the sound wavepasses through it. The motion of theparticles parallel (and anti-parallel) to thedirection of the energy transport is whatcharacterizes sound as a longitudinal wave.

A vibrating tuning forkis capable of creatingsuch a longitudinal

wave. As the tines of the fork vibrate backand forth, they push on neighboring airparticles. The forward motion of a tinepushes air molecules horizontally to theright and the backward retraction of thetine creates a low pressure area allowingthe air particles to move back to the left.Because of the longitudinal motion of theair particles, there are regions in the air

where the air particles are compressedtogether and other regions where the airparticles are spread apart. These regionsare known as compressions andrarefactions respectively. Thecompressions are regions of high airpressure while the rarefactions are regionsof low air pressure. The diagram belowdepicts a sound wave created by a tuningfork and propagated through the air in an

open tube. The compressions andrarefactions are labeled.





The wavelength of a wave is merely thedistance which a disturbance travels alongthe medium in one complete wave cycle.Since a wave repeats its pattern once everywave cycle, the wavelength is sometimesreferred to as the length of the repeatingpattern - the length of one complete wave.For a transverse wave, this length iscommonly measured from one wave crest

to the next adjacent wave crest, or fromone wave trough to the next adjacent wavetrough. Since a longitudinal wave does notcontain crests and troughs, its wavelengthmust be measured differently. Alongitudinal wave consists of a repeatingpattern of compressions and rarefactions.Thus, the wavelength is commonlymeasured as the distance from onecompression to the next adjacent

compression or the distance from onerarefaction to the next adjacent rarefaction.

Since a sound wave consists of a repeatingpattern of high pressure and low pressureregions moving through a medium, it issometimes referred to as a pressurewave. If a detector, whether it be thehuman ear or a man-made instrument, isused to detect a sound wave, it would

detect fluctuations in pressure as the soundwave impinges upon the detecting device.At one instant in time, the detector woulddetect a high pressure; this wouldcorrespond to the arrival of a compressionat the detector site. At the next instant intime, the detector might detect normalpressure. And then finally a low pressurewould be detected, corresponding to thearrival of a rarefaction at the detector site.

Since the fluctuations in pressure asdetected by the detector occur at periodicand regular time intervals, a plot of pressure vs. time would appear as a sine




curve. The crests of the sine curvecorrespond to compressions; the troughscorrespond to rarefactions; and the "zeropoint" corresponds to the pressure which

the air would have if there were nodisturbance moving through it. The diagrambelow depicts the correspondence betweenthe longitudinal nature of a sound wave andthe pressure-time fluctuations which itcreates.

The above diagram can be somewhatmisleading if you are not careful. Therepresentation of sound by a sine wave ismerely an attempt to illustrate the

sinusoidal nature of the pressure-timefluctuations. Do not conclude that sound isa transverse wave which has crests andtroughs. Sound is indeed a longitudinalwave with compressions and rarefactions.As sound passes through a medium, theparticles of that medium do not vibrate in atransverse manner. Do not be misled -sound is a longitudinal wave.

Check Your Understanding

1. A sound wave is a pressure wave;regions of high (compressions) and lowpressure (rarefactions) are established asthe result of the vibrations of the soundsource. These compressions and

rarefactions result because sound

a. is more dense than air and thus hasmore inertia, causing the bunching up

2001 The Physics Classroom and Mathsoft Education and




PhysicsTutorial


Lesson 1

Lesson 2

Pitch andFrequency

Intensityand theDecibel

ScaleThe Speedof Sound

TheHuman Ear

Lesson 3

Lesson 4

Lesson 5

Lesson 2: Sound Properties and Their

Perception

Pitch and Frequency

A sound wave, like any other wave, isintroduced into a medium by a vibratingobject. The vibrating object is the source of the disturbance which moves through the

medium. The vibrating object which createsthe disturbance could be the vocal chords of a person, the vibrating string and soundboard of a guitar or violin, the vibratingtines of a tuning fork, or the vibratingdiaphragm of a radio speaker. Regardless of what vibrating object is creating the soundwave, the particles of the medium throughwhich the sound moves is vibrating in aback and forth motion at a given

frequency. The frequency of a wave refersto how often the particles of the mediumvibrate when a wave passes through themedium. The frequency of a wave ismeasured as the number of complete back-and-forth vibrations of a particle of themedium per unit of time. If a particle of airundergoes 1000 longitudinal vibrations in 2seconds, then the frequency of the wavewould be 500 vibrations per second. A

commonly used unit for frequency is theHertz (abbrviated Hz), where

1 Hertz = 1 vibration/second

As a sound wave moves through a medium,each particle of the medium vibrates at thesame frequency. This is sensible since eachparticle vibrates due to the motion of itsnearest neighbor. The first particle of the

medium begins vibrating, at say 500 Hz,and begins to set the second particle intovibrational motion at the same frequency of 500 Hz. The second particle begins




vibrating at 500 Hz and thus sets the thirdparticle of the medium into vibrationalmotion at 500 Hz. The process continuesthroughout the medium; each particle

vibrates at the same frequency. And of course the frequency at which each particlevibrates is the same as the frequency of theoriginal source of the sound wave.Subsequently, a guitar string vibrating at500 Hz will set the air particles in the roomvibrating at the same frequency of 500 Hzwhich carries a sound signal to the ear of alistener which is detected as a 500 Hzsound wave.

The back-and-forth vibrational motion of the particles of the medium would not bethe only observable phenomenon occurringat a given frequency. Since a sound wave isa pressure wave, a detector could be usedto detect oscillations in pressure from ahigh pressure to a low pressure and back toa high pressure. As the compression (highpressure) and rarefaction (low pressure)

disturbances move through the medium,they would reach the detector at a givenfrequency. For example, a compressionwould reach the detector 500 times persecond if the frequency of the wave were500 Hz. Similarly, a rarefaction would reachthe detector 500 times per second if thefrequency of the wave were 500 Hz. Thusthe frequency of a sound wave not onlyrefers to the number of back-and-forth

vibrations of the particles per unit of time,but also refers to the number of compression or rarefaction disturbanceswhich pass a given point per unit of time. Adetector could be used to detect thefrequency of these pressure oscillationsover a given period of time. The typicaloutput provided by such a detector is apressure-time plot as shown below.







Since a pressure-time plot shows thefluctuations in pressure over time, theperiod of the sound wave can be found bymeasuring the time between successivehigh pressure points (corresponding to thecompressions) or the time betweensuccessive low pressure points

(corresponding to the rarefactions). Asdiscussed in an earlier unit, the frequencyis simply the reciprocal of the period. Forthis reason, a sound wave with a highfrequency would correspond to a pressuretime plot with a small period - that is, a plotcorresponding to a small amount of timebetween successive high pressure points.Conversely, a sound wave with a lowfrequency would correspond to a pressure

time plot with a large period - that is, a plotcorresponding to a large amount of timebetween successive high pressure points.The diagram below shows two pressure-time plots,one corresponding to a highfrequency and the other to a low frequency.

The ears of humans (and other animals) aresensitive detectors capable of detecting thefluctuations in air pressure which impinge

upon the eardrum. The mechanics of theear's detection ability will be discussed laterin this lesson. For now, it is sufficient to saythat the human ear is capable of detecting




sound waves with with a wide range of frequencies, ranging betweenapproximately 20 Hz to 20 000 Hz. Anysound with a frequency below the audible

range of hearing (i.e., less than 20 Hz) isknown as an infrasound and any soundwith a frequency above the audible range of hearing (i.e., more than 20 000 Hz) isknown as an ultrasound. Humans are notalone in their ability to detect a wide rangeof frequencies. Dogs can detect frequenciesas low as approximately 50 Hz and as highas 45 000 Hz. Cats can detect frequenciesas low as approximately 45 Hz and as high

as 85 000 Hz. Bats, who are essentiallyblind and must rely on sound echolation fornavigation and hunting, can detectfrequecies as high as 120 000 Hz. Dolphinscan detect frequencies as high as 200 000Hz. While dogs, cats, bats, and dolphinshave an unusual ability to detectultrasound, an elephant possesses theunusual ability to detect infrasound, havingan audible range from approximately 5 Hz

to approxmately 10 000 Hz.

The sensations of these frequencies arecommonly referred to as the pitch of asound. A high pitch sound corresponds to ahigh frequency and a low pitch soundcorresponds to a low frequency. Amazingly,many people, especially those who haebeen musically trained, are capable of detecting a difference in frequency between

two separate sounds which is as little as 2Hz. When two sounds with a frequencydifference of greater than 7 Hz are playedsimultaneously, most people are capable of detecting the presence of a complex wavepattern resulting from the interference andsuperposition of the two sound waves.Certain sound waves when played (andheard) simultaneously will produce aparticularly pleasant sensation when heard,

are are said to be consonant. Such soundwaves form the basis of intervals in music.For example, any two sounds whosefrequencies make a 2:1 ratio are said to be




separated by an octave and result in aparticularly pleasing sensation when heard;that is, two sound waves sound good whenplayed together if one sound has twice the

frequency of the other. Similarly twosounds with a frequency ratio of 5:4 aresaid to be separated by an interval of athird; such sound waves also sound goodwhen played together. Examples of othersound wave intervals and their respectivefrequency ratios are listed in the tablebelow.

Interval

Frequency

Ratio Examples

Octave 2:1512 Hz and

256 Hz

Third 5:4320 Hz and

256 Hz

Fourth 4:3342 Hz and

256 Hz

Fifth 3:2384 Hz and

256 Hz

The ability of humans to perceive pitch isassociated with the frequency of the soundwave which impinges upon the ear.Because sound waves are longitudinalwaves which produce high- and low-pressure disturbances of the particles of a

medium at a given frequency, the ear hasan ability to detect such frequencies andassociate them with the pitch of the sound.But pitch is not the only property of a soundwave detectable by the human ear. In thenext part of Lesson 2, we will investigatethe ability of the ear to perceive theintensity of a sound wave.


1. Two notes which have a frequency ratioof 2:1 are said to be separated by an




PhysicsTutorial


Lesson 1

Lesson 2

Pitch andFrequency



TheHuman Ear

Lesson 3

Lesson 4

Lesson 5


Perception

Intensity and the Decibel Scale

A sound wave is introduced into a mediumby the vibration of an object. For example, avibrating guitar string forces surrounding airmolecules to be compressed and expanded,

creating a pressure disturbance consisting of an alternating pattern of compressions andrarefactions. Thedisturbance then travelsfrom particle to particlethrough the medium,transporting energy as it moves. The energywhich is carried by the disturbance wasoriginally imparted to the medium by the

vibrating string. The amount of energy whichis transferred to the medium is dependentupon the amplitude of vibrations of theguitar string. If the more energy is put intothe plucking of the string (that is, more workis done to displace the string a greateramount from its rest position), then thestring vibrates with a wider amplitude. Thegreater amplitude of vibration of the guitarstring thus imparts more energy to the

medium, causing air particles to bedisplaced a greater distance from their restposition. Subsequently, the amplitude of vibration of the particles of the medium isincreased, corresponding to an increasedamount of energy being carried by theparticles. This relationship between energyand amplitude was discussed in more detailin a previous unit.

The amount of energy which is transportedpast a given area of the medium per unit of time is known as the intensity of the soundwave. The greater the amplitude of




vibrations of the particles of the medium,the greater the rate at which energy istransported through it, and the more intensethat the sound wave is. Intensity is the

energy/time/area; and since the energy/time ratio is equivalent to the quantitypower, intensity is simply the power/area.

Typical units for expressing the intensity of a

sound wave are Watts/meter2.

As a sound wave carries its energy through

a two-dimensional or three-dimensionalmedium, the intensity of the sound wavedecreases with increasing distance from thesource. The decrease inintensity with increasingdistance is explained by thefact that the wave isspreading out over a circular(2 dimensions) or spherical (3dimensions) surface and thus

the energy of the sound waveis being distributed over a greater surfacearea. The diagram at the right shows thatthe sound wave in a 2-dimensional mediumis spreading out in space over a circularpattern. Since energy is conserved and thearea through which this energy istransported is increasing, the power (being aquantity which is measured on a per areabasis) must decrease. The mathematical

relationship between intensity and distanceis sometimes referred to as an inversesquare relationship. As the intensityvaries inversely with the square of thedistance from the source. So if the distancefrom the source is doubled (increased by afactor of 2), then the intensity is quartered(decreased by a factor of 4). Similarly, if thedistance from the source is quadrupled, thenthe intensity is decreased by a factor of 16.

Applied to the diagram at the right, theintensity at point B is one-fourth theintensity as point A and the intensity atpoint C is one-sixteenth the intensity at




point A. Since the intensity-distancerelationship is an inverse relationship, anincrease in one quantity corresponds to adecrease in the other quantity. And since

the intensity-distance relationship is aninverse square relationship, whatever factorby which the distance is increased, theintensity is decreased by a factor equal tothe square of the "distance change factor."The sample data in the table below illustratethe inverse square relationship betweenpower and distance.

Distance Intensity

1 m 160 units

2 m 40 units

3 m 17.8 units

4 m 10 units

Humans are equipped with very sensitiveears capable of detecting sound waves of

extremely low intensity. The faintest soundwhich the typical human ear can detect has

an intensity of 1*10-12 W/m2. This intensitycorresponds to a pressure wave in which acompression of the particles of the mediumincreases the air pressure in thatcompressional region by a mere 0.3billionths of an atmosphere. A sound with an

intensity of 1*10-12 W/m2 corresponds to asound which will displace particles of air by a

mere one-billionth of a centimeter. Thehuman ear can detect such a sound. WOW!This faintest sound which the human ear candetect is known as the threshold of hearing. The most intense sound which theear can safely detect without suffering anyphysical damage is more than one billiontimes more intense than the threshold of hearing.

Since the range of intensities which thehuman ear can detect is so large, the scalewhich is frequently used by physicists tomeasure intensity is a scale based on




multiples of 10. This type of scale issometimes referred to as a logarithmicscale. The scale for measuring intensity isthe decibel scale. The threshold of hearing

is assigned a sound level of 0 decibels(abbreviated 0 dB); this sound corresponds

to an intensity of 1*10-12 W/m2. A sound

which is 10 times more intense ( 1*10-11

W/m2) is assigned a sound level of 10 dB. Asound which is 10*10 or 100 times more

intense ( 1*10-10 W/m2) is assigned asound level of 20 db. A sound which is10*10*10 or 1000 times more intense (

1*10-9 W/m2) is assigned a sound level of 30 db. A sound which is 10*10*10*10 or

10000 times more intense ( 1*10-8 W/m2)is assigned a sound level of 40 db. Observethat this scale is based on powers or

multiples of 10. If one sound is 10x timesmore intense than another sound, then ithas a sound level which is 10*x moredecibels than the less intense sound. Thetable below lists some common sounds with

an estimate of their intensity and decibellevel.

Source IntensityIntensity

Level

#Times

GreaterThanTOH

Threshold of

Hearing(TOH)

1*10-12

W/m2 0 dB 100

RustlingLeaves

1*10-11

W/m210 dB 101

Whisper1*10-10

W/m220 dB 102

NormalConversation

1*10-6

W/m260 dB 106

Busy StreetTraffic

1*10-5

W/m270 dB 107




VacuumCleaner

1*10-4

W/m280 dB 108

Large

Orchestra

6.3*10-3

W/m2

98 dB 109.8

Walkman atMaximum

Level

1*10-2

W/m2100 dB 1010

Front Rows of Rock Concert

1*10-1

W/m2110 dB 1011

Threshold of Pain

1*101 W/

m2130 dB 1013

Military JetTakeoff

1*102 W/m2

140 dB 1014

InstantPerforation of

Eardrum

1*104 W/

m2160 dB 1016

While the intensity of a sound is a veryobjective quantity which can be measured

with sensitive instrumentation, theloudness of a sound is more of a subjectiveresponse which will vary with a number of factors. The same sound will not beperceived to have the same loudness to allindividuals. Age is one factor which effectsthe human ear's response to a sound. Quiteobviously, your grandparents do not hearlike they used to. The same intensity soundwould not be perceived to have the same

loudness to them as it would to you.Furthermore, two sounds with the sameintensity but different frequencies will not beperceived to have the same loudness.Because of the human ear's tendency toamplify sounds having frequencies in therange from 1000 Hz to 5000 Hz, sounds withthese intensities seem louder to the humanear. Despite the distinction betweenintensity and loudness, it is safe to state

that the more intense sounds will beperceived to be the loudest sounds.




PhysicsTutorial


Lesson 1

Lesson 2

Pitch andFrequency



TheHuman Ear

Lesson 3

Lesson 4

Lesson 5


Perception

The Speed of Sound

A sound wave is a pressure disturbancewhich travels through a medium by meansof particle interaction. As one particlebecomes disturbed, it exerts a force on the

next adjacent particle, thus disturbing thatparticle from rest and transporting theenergy through the medium. Like anywave, the speed of a sound wave refers tohow fast the disturbance is passed fromparticle to particle. While frequency refersto the number of vibrations which anindividual particle makes per unit of time,speed refers to the distance which thedisturbance travels per unit of time. Always

be cautious to distinguish between the twooften confused quantities of speed (how fast...) and frequency (how often...).

Since the speed of a wave is defined as thedistance which a point on a wave (such as acompression or a rarefaction) travels perunit of time, it is often expressed in units of meters/second (abbreviated m/s). Inequation form, this is

speed = distance/time

The faster which a sound wave travels, themore distance it will cover in the sameperiod of time. If a sound wave is observedto travel a distance of 700 meters in 2seconds, then the speed of the wave wouldbe 350 m/s. A slower wave would coverless distance - perhaps 600 meters - in thesame time period of 2 seconds and thushave a speed of 300 m/s. Faster wavescover more distance in the same period of time.




The speed of any wave depends upon theproperties of the medium through which thewave is traveling. Typically there are two

essential types of properties which effectwave speed - inertial properties and elasticproperties. The density of a medium is anexample of an inertial property. Thegreater the inertia (i.e., mass density) of individual particles of the medium, the lessresponsive they will be to the interactionsbetween neighboring particles and theslower the wave. If all other factors areequal (and seldom is it that simple), a

sound wave will travel faster in a less densematerial than a more dense material. Thus,a sound wave will travel nearly three timesfaster in Helium as it will in air; this ismostly due to the lower mass of Heliumparticles as compared to air particles.

Elastic properties are those propertiesrelated to the tendency of a material toeither maintain its shape and not deformwhenever a force or stress is applied to it. Amaterial such as steel will experience avery small deformation of shape (anddimension) when a stress is applied to it.Steel is a rigid material with a highelasticity. On the other hand, a materialsuch as a rubber band is highly flexible;when a force is applied to stretch therubber band, it deforms or changes itsshape readily. A small stress on the rubber

band causes a large deformation. Steel isconsidered to be a stiff or rigid material,whereas a rubber band is considered aflexible material. At the particle level, a stiff or rigid material is characterized by atomsand/or molecules with strong attractions foreach other. When a force is applied in anattempt to stretch or deform the material,its strong particle interactions prevent thisdeformation and help the material maintain

its shape. Rigid materials such as steel areconsidered to have a high elasticity (elasticmodulus is the technical term). The phaseof matter has a tremendous impact upon




the elastic properties of the medium. Ingeneral, solids have the strongestinteractions between particles, followed byliquids and then gases. For this reason,

longitudinal sound waves travel faster insolids than they do in liquids than they doin gases. Even though the inertial factormay favor gases, the elastic factor has agreater influence on the speed (v) of awave, thus yielding this general pattern:

vsolids > vliquids > vgases

The speed of a sound wave in air dependsupon the properties of the air, namely the

temperature and the pressure. Thepressure of air (like any gas) will effect themass density of the air (an inertialproperty) and the temperature will effectthe strength of the particle interactions (anelastic property). At normal atmosphericpressure, the temperature dependence of the speed of a sound wave through air isapproximated by the following equation:

v = 331 m/s + (0.6 m/s/C)*T

where T is the temperature of the air indegrees Celsius. Using this equation is usedto determine the speed of a sound wave inair at a temperature of 20 degrees Celsiusyields the following solution.

v = 331 m/s + (0.6 m/s/C)*T

v = 331 m/s + (0.6 m/s/C)*20 C

v = 331 m/s + 12 m/s

v = 343 m/s

At normal atmospheric pressure and atemperature of 20 degrees Celsius, a soundwave will travel at approximately 343 m/s;this is approximately equal to 750 miles/

hour. While this speed may seem fast byhuman standards (the fastest humans cansprint at approximately 11 m/s and highwayspeeds are approximately 30 m/s), the




speed of a sound wave is slow incomparison to the speed of a light wave.Light travels through air at a speed of approximately 300 000 000 m/s; this is

nearly 900 000 times the speed of sound.For this reason, humans can observe adetectable time delay between the thunderand lightning during a storm. The arrival of the light wave from the location of thelightning strike occurs in so little time thatit is essentially negligible. Yet the arrival of the sound wave from the location of thelightning strike occurs much later. The timedelay between the arrival of the light wave

(lightning) and the arrival of the soundwave (thunder) allows a person toapproximate his/her distance from thestorm location. For instance if the thunderis heard 3 seconds after the lightning isseen, then sound (whose speed isapproximated as 345 m/s) has traveled adistance of

distance = v * t = 345 m/s * 3 s = 1035

m

If this value is converted to miles (divide by1600 m/1 mi), then the storm is a distanceof 0.65 miles away.

Another phenomenon related to theperception of time delays between twoevents is the phenomenon of echolation. A

person can often perceive a time delaybetween the production of a sound and thearrival of a reflection of that sound off adistant barrier. If you have ever made aholler within a canyon, perhaps you haveheard an echo of your holler off a distantcanyon wall. The time delay between theholler and the echo corresponds to the timefor the holler to travel the round-tripdistance to the canyon wall and back. A

measurement of this time would allow aperson to estimate the one-way distance tothe canyon wall. For instance if an echo isheard 2.2 seconds after making the holler ,




then the distance to the canyon wall can befound as follows:

distance = v * t = 345 m/s * 1.1 s = 380

mThe canyon wall is 380 meters away. Youmight have noticed that the time of 1.1seconds is used in the equation. Since thetime delay corresponds to the time for theholler to travel the round-trip distance tothe canyon wall and back, the one-waydistance to the canyon wall corresponds toone-half the time delay.

While the phenomenon of echolation is of relatively minimal importance to humans, itis an essential trick of the trade for bats.Being merely blind, bats must use soundwaves to navigate and hunt. They produceshort bursts of ultrasonic sound waveswhich reflect off their surroundings andreturn. Their detection of the time delaybetween the sending and receiving of thepulses allows a bat to approximate the

distance to surrounding objects. Some bats,known as Doppler bats, are capable of detecting the speed and direction of anymoving objects by monitoring the changesin frequency of the reflected pulses. Thesebats are utilizing the physics of the Dopplereffect discussed in an earlier unit (and alsoto be discussed later in Lesson 3). Thismethod of echolation enables a bat tonavigate and to hunt.

Like any wave, a sound wave has a speedwhich is mathematically related to thefrequency and the wavelength of the wave.As discussed in a previous unit, themathematical relationship between speed,frequency and wavelength is given be thefollowing equation.

Speed = Wavelength * Frequency

Using the symbols v, , and f , the equationcan be re-written as

v = f *




The above equations are useful for solvingmathematical problems related to thespeed, frequency and wavelength

relationship. However, one importantmisconception could be conveyed by theequation. Even though wave speed iscalculated using the frequency and thewavelength, the wave speed is notdependent upon these quantities. Analteration in wavelength does not effect(i.e., change) wave speed. Rather, analteration in wavelength effects thefrequency in an inverse manner. A doubling

of the wavelength results in a halving of thefrequency; yet the wave speed is notchanged. The speed of a sound wavedepends on the properties of the mediumthrough which it moves and the only way tochange the speed is to change theproperties of the medium.


1. An automatic focus camera is able tofocus on objects by use of an ultrasonicsound wave. The camera sends out soundwaves which reflect off distant objects andreturn to the camera. A sensor detects thetime it takes for the waves to return andthen determines the distance an object is

from the camera. If a sound wave (speed =340 m/s) returns to the camera 0.150seconds after leaving the camera, how faraway is the object?

2. The annoying sound from a mosquito isproduced when it beats its wings at theaverage rate of 600 wingbeats per second.

a. What is the frequency in Hertz of thesound wave?




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Lesson 1

Lesson 2

Pitch andFrequency

Intensityand the

DecibelScale

The Speedof Sound

TheHuman Ear

Lesson 3

Lesson 4


Perception

The Human Ear

Understanding how humans hear is acomplex subject involving the fields of physiology, psychology and acoustics. Inthis part of Lesson 2, we will focus on the

acoustics (the branch of physics pertainingto sound) of hearing. We will attempt tounderstand how the human ear serves asan astounding transducer, converting soundenergy to mechanical energy to a nerveimpulse which is transmitted to the brain.The ear's ability to do this allows us toperceive the pitch of sounds by detection of the wave's frequencies, the loudness of sound by detection of the wave's amplitude

and the timbre of the sound by thedetection of the various frequencies whichmake up a complex sound wave.

The ear consists of three basic parts - theouter ear, the middle ear, and the innerear. Each part of the ear serves a specificpurpose in the task of detecting andinterpreting sound. The outer ear serves tocollect and channel sound to the middle

ear. The middle ear serves to transform theenergy of a sound wave into the internalvibrations of the bone structure of themiddle ear and ultimately transform thesevibrations into a compressional wave in theinner ear. The inner ear serves to transformthe energy of a compressional wave withinthe inner ear fluid into nerve impulseswhich can be transmitted to the brain. Thethree parts of the ear are shown below.






The outer ear consists of an ear flap and anapproximately 2-cm long ear canal. The earflap provides protection for the middle earin order to prevent damage to the eardrum.

The outer ear also channels sound waveswhich reach the ear through the ear canalto the eardrum of the middle ear. Becauseof the length of the ear canal, it is capableof amplifying sounds with frequencies of approximately 3000 Hz. As sound travelsthrough the outer ear, the sound is still inthe form of a pressure wave, with analternating pattern of high and low pressureregions. It is not until the sound reachesthe eardrum at the interface of the outerand the middle ear that the energy of themechanical wave becomes converted intovibrations of the inner bone structure of theear.

The middle ear is an air-filled cavity whichconsists of an eardrum and three tiny,interconnected bones - thehammer, anvil, and stirrup.

The eardrum is a verydurable and tightlystretched membrane whichvibrates as the incoming pressure wavesreach it. As shown at the right, acompression forces the eardrum inward anda rarefaction forces the eardrum outward,thus vibrating the eardrum at the samefrequency of the sound wave. Beingconnected to the hammer, the movementsof the eardrum will set the hammer, anvil,and stirrup into motion at the samefrequency of the sound wave. The stirrup is





connected to the inner ear; and thus thevibrations of the stirrup are transmitted tothe fluid of the middle ear and create acompression wave within the fluid. The

three tiny bones of the middle ear act aslevers to amplify the vibrations of thesound wave. Due to a mechanicaladvantage, the displacements of the stirrupare greater than that of the hammer.Furthermore, since the pressure wavestriking the large area of the eardrum isconcentrated into the smaller area of thestirrup, the force of the vibrating stirrup isnearly 15 times larger than that of the

eardrum. This feature enhances our abilityof hear the faintest of sounds. The middleear is an air-filled cavity which is connectedby the Eustachian tube to the mouth. Thisconnection allows for the equalization of pressure within the air-filled cavities of theear. When this tube becomes cloggedduring a cold, the ear cavity is unable toequalize its pressure; this will often lead toearaches and other pains.

The inner ear consists of a cochlea, thesemicircular canals, and the auditory nerve.The cochlea and the semicircular canals arefilled with a water-like fluid. The fluid andnerve cells of the semicircular canalsprovide no roll in the task of hearing; theymerely serve as accelerometers fordetecting accelerated movements andassisting in the task of maintaining balance.

The cochlea is a snail-shaped organ whichwould stretch to approximately 3 cm. Inaddition to being filled with fluid, the innersurface of the cochlea is lined with over 20000 hair-like nerve cells which perform oneof the most critical roles in our ability tohear. These nerve cells have a differ inlength by minuscule amounts; they alsohave different degrees of resiliency to thefluid which passes over them. As a

compressional wave moves from theinterface between the hammer of themiddle ear and the oval window of the innerear through the cochlea, the small hair-like




nerve cells will be set in motion. Each haircell has a natural sensitivity to a particularfrequency of vibration. When the frequencyof the compressional wave matches the

natural frequency of the nerve cell, thatnerve cell will resonate with a largeramplitude of vibration. This increasedvibrational amplitude induces the cell torelease an electrical impulse which passesalong the auditory nerve towards the brain.In a process which is not clearlyunderstood, the brain is capable of interpreting the qualities of the sound uponreception of these electric nerve impulses.

Go to Lesson 3




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Lesson 1

Lesson 2

Lesson 3

Interferenceand Beats

The DopplerEffect and

ShockWaves

BoundaryBehaviorReflection,Refraction,andDiffraction

Lesson 4

Lesson 5


Interference and Beats

Wave interference is the phenomenonwhich occurs when two waves meet whiletraveling along the same medium. Theinterference of waves causes the medium totake on a shape which results from the net

effect of the two individual waves upon theparticles of the medium. As mentioned in aprevious unit of The Physics Classroom, if two crests having the same shape meet upwith one another while traveling in oppositedirections along a medium, the medium willtake on the shape of a crest with twice theamplitude of the two interfering crests. Thistype of interference is known asconstructive interference. If a crest and

a trough having the same shape meet upwith one another while traveling in oppositedirections along a medium, the two pulseswill cancel each other's effect upon thedisplacement of the medium and themedium will assume the equilibriumposition. This type of interference is knownas destructive interference. Thediagrams below show two waves - one isblue and the other is red - interfering in

such a way to produce a resultant shape ina medium; the resultant is shown in green.In two cases (on the left and in themiddle), constructive interference occursand in the third case (on the far right,destructive interference occurs.

But how can sound waves which do notpossess crests and troughs interfere




constructively and destructively? Sound is apressure wave which consists ofcompressions and rarefactions. As acompression passes through a section of a

medium, it tends to pull particles togetherinto a small region of space, thus creating ahigh pressure region. And as a rarefactionpasses through a section of a medium, ittends to push particles apart, thus creatinga low pressure region. The interference of sound waves causes the particles of themedium to behave in a manner that reflectsthe net effect of the two individual wavesupon the particles. For example, if a

compression (high pressure) of one wavemeets up with a compression (highpressure) of a second wave at the samelocation in the medium, then the net effectis that that particular location willexperience an even greater pressure. Thisis a form of constructive interference. If tworarefactions (two low pressuredisturbances) from two different soundwaves meet up at the same location, then

the net effect is that that particular locationwill experience an even lower pressure.This is also an example of constructiveinterference. Now if a particular locationalong the medium repeatedly experiencesthe interference of two compressionsfollowed up by the interference of tworarefactions, then the two sound waves willcontinually reinforce each other andproduce a very loud sound. The loudness of

the sound is the result of the particles atthat location of the medium undergoingoscillations from very high to very lowpressures. As mentioned in a previous unit,locations along the medium whereconstructive interference continually occursare known as anti-nodes. The animationbelow shows two sound waves interferingconstrucively in order to produce very largeoscillations in pressure at a variety of anti-

nodal locations. Note that compressions arelabeled with a C and rarefactions arelabeled with an R.




Now if two sound waves interfere at a given

location in such a way that the compressionof one wave meets up with the rarefactionof a second wave, destructive interferenceresults. The net effect of a compression(which pushes particles together) and ararefaction (which pulls particles apart)upon the particles in a given region of themedium is to not even cause adisplacement of the particles. The tendencyof the compression to push particles

together is cancelled by the tendency of therarefactions to pull particles apart; theparticles would remain at their rest positionas though there wasn't even a disturbancepassing through them. This is a form of destructive interference. Now if a particularlocation along the medium repeatedlyexperiences the interference of acompression and rarefaction followed up bythe interference of a rarefaction and a

compression, then the two sound waves willcontinually cancel each other and no soundis heard. The absence of sound is the resultof the particles remaining at rest andbehaving as though there were nodisturbance passing through it. Amazingly,in a situation such as this, two sound waveswould combine to produce no sound. Asmentioned in a previous unit, locationsalong the medium where destructive

interference continually occurs are knownas nodes.

You might remember the popular classroomdemonstration involving the interference of two sound waves from two speakers. Thespeakers were set approximately 1 meterapart and produced identical tones. The twosound waves traveled through the air infront of the speakers, spreading our

through the room in spherical fashion. Asnapshot in time of the appearance of thesewaves is shown in the diagram below. Inthe diagram, the compressions of a




wavefront are represented by a thick lineand the rarefactions are represented bythin lines. These two waves interfere insuch a manner as to produce locations of

some loud sounds and other locations of nosound. Of course the loud sounds are heardat locations where ccompressions meetcompressions or rarefactions meetrarefactions and the "no sound" locationsappear wherever the compressions of oneof the waves meet the rarefactions of theother wave. If you were to plug one ear andturn the other ear towards the place of thespeakers and then slowly walk across the

room parallel to the plane of the speakers,then you would encounter an amazingphenomenon. You would alternatively hearloud sounds as you approached anti-nodallocations and virtually no sound as youapproached nodal locations. (As you mayhave observed, the nodal locations werenot true nodal locations due to reflections of sound waves off the walls which tended tofill the entire room with reflected sound.

Even though the sound waves whichreached the nodal locations directly fromthe speakers destructively interfered, otherwaves reflecting off the walls tended toreach that same location to produce apressure disturbance.)

Destructive interference of sound wavesbecomes an important issue in the design of concert halls and auditoriums. The roomsmust be designed in such as way as toreduce the amount of destructive

interference. Interference can occur as theresult of sound from two speakers meetingat the same location as well as the result of sound from a speaker meeting with sound




reflected off the walls and ceilings. If thesound arrives at a given location such thatcompressions meet rarefactions, thendestructive interference will occur resulting

in a reduction in the loudness of the soundat that location. One means of reducing theseverity of destructive interference is by thedesign of walls, ceilings, and baffles thatserve to absorb sound rather than reflect it.This will be discussed in more detail later inLesson 3.

The destructive interference of sound wavescan also be used for advantageously in

noise reduction systems. Ear phoneshave been produced which can be used byfactory and construction workers to reducethe noise levels on their jobs. Such earphones capture sound from theenvironment and use computer technologyto produce a second sound wave whichone-half cycle out of phase. Thecombination of these two sound waveswithin the headset will result in destructive

interference and thus reduce a worker'sexposure to loud noise.

Interference of sound waves haswidespread applications in the world of music. Music seldom consists of soundwaves of a single frequency playedcontinuously. Few music enthusiasts wouldbe impressed by an orchestra which playedmusic consisting of the note with a pure

tone played by all instruments in theorchestra. Hearing a sound wave of 256 Hz(middle C) would become rathermonotonous (both literally and figuratively).Rather, instruments are known to produceovertones when played resulting in a soundwhich consists of a multiple of frequencies.Such instruments are described as beingrich in tone color. And even the best choirswill earn their money when two singers sing

two notes (i.e., produce two sound waves)which are an octave apart. Music is amixture of sound waves which typicallyhave whole number ratios between the




frequencies associated with their notes. Infact, the major distinction between musicand noise is that noise consists of a mixtureof frequencies which have no mathematical

order to them and music consists of amixture of frequencies which have a clearmathematical relationship between them.While it may be true that "one person'smusic is another person's noise" (e.g., yourmusic might be thought of by your parentsas being noise), a physical analysis of musical sounds reveals a mixture of soundwaves which are mathematically related.

To demonstrate this nature of music, let'sconsider one of the simplest mixtures of two different sound waves - two soundwaves with a 2:1 frequency ratio. Thiscombination of waves is known as anoctave. A simple sinusoidal plot of the wavepattern for two such waves is shown below.Note that the red wave has two times thefrequency of the blue wave. Also observethat the interference of these two waves

produces a resultant (in green) which has aperiodic and repeating pattern. One mightsay that two sound waves which have aclear whole number ratio between theirfrequencies interfere to produce a wavewith a regular and repeating pattern; theresult is music.

Another simple example of two soundwaves with a clear mathematicalrelationship between frequencies is shownbelow. Note that the red wave has three-halves the frequency of the blue wave. Inthe music world, such waves are said to bea fifth apart and represent a popular

musical interval. Observe once more thatthe interference of these two wavesproduces a resultant (in green) which has aperiodic and repeating pattern. It should be




said again: two sound waves which have aclear whole number ratio between theirfrequencies interfere to produce a wavewith a regular and repeating pattern; the

result is music.

Finally, the diagram below illustrates anexample of noise. The diagram shows twowaves interfering, but this time there is noclear mathematical relationship betweentheir frequencies (in computer terms, onehas a wavelength of 37 and the other has awavelength 20 pixels). Observe (lookcarefully) that the pattern of the resultantis neither periodic nor repeating. Themessage is clear: if two sound waves whichhave no simple mathematical relationshipbetween their frequencies interfere toproduce a wave, the result will be anirregular and non-repeating pattern; this is"noise."

A final application of music to the world of physics pertains to the topic of beats.Beats are the periodic and repeatingfluctuations heard in the intensity of asound when two sound waves of verysimilar frequencies interfere with oneanother. The diagram below illustrates thewave interference pattern resulting fromtwo waves (drawn in red and blue) with

very similar frequencies. A beat pattern ischaracterized by a wave whose amplitude ischanging at a regular rate. Observe thatthe beat pattern (drawn in green)




repeatedly oscillates from zero amplitude toa large amplitude, back to zero amplitudethroughout the pattern. Points of constructive interference (C.I.) and

destructive interference (D.I.) are labeledon the diagram. When constructiveinterference occurs, a loud sound is heard;this corresponds to a peak on the beatpattern (drawn in green). When destructiveinterference occurs, no sound is heard; thiscorresponds to a point of no displacementon the beat pattern. Since there is a clearrealtionship between the amplitude and theloudness, this beat pattern would be

consistent with a wave which varies involume at a regular rate.

The beat frequency refers to the rate atwhich the volume is heard to be oscillatingfrom high to low volume. For example, if two complete cycles of high and lowvolumes are heard every second, the beatfrequency is 2 Hz. The beat frequency isalways equal to the difference in frequencyof the two notes which interfere to producethe beats. So if two sound waves withfrequencies of 256 Hz and 254 Hz areplayed simultaneously, a beat frequency of 2 Hz will be detected. Beats were producedin a classroom demonstration using twotuning forks. Though the tuning forks wereidentical, the frequency of one of the forkswas lowered by wrapping one of the tineswith a rubber bands. The result was thatthe two tuning forks produced sounds withslightly different frequencies which




interfered to produce detectable beats. Thehuman ear is capable of detecting beatswith frequencies of 7 Hz and below.

A piano tuner frequently utilizes thephenomenon of beats to tune a pianostring. She will pluck the string and tap atuning fork at the same time. If the twosound sources - the piano string and thetunng fork - produce detectable beats thentheir frequencies are not identical. She willthen adjust the tension of the piano stringand repeat the process until the beats canno longer be heard. As the piano string

becomes more in tune with the tuning fork,the beat frequency will be reduced andapproach 0 Hz. When beats are no longerheard, the piano string is tuned to thetuning fork; that is, they play the samefrequency.

Important Note: Many of the diagrams onthis page represent a sound wave by a sinewave. Such a wave more closely resemblesa transverse wave and may mislead peopleinto thinking that sound is a transversewave. Sound is not a transverse wave, butrather a longitudinal wave. Nonetheless,the variations in pressure with time take onthe pattern of a sine wave and thus a sinewave is often used to represent thepressure-time features of a sound wave.

Check Your UnderstandingTwo speakers arearranged so thatsound waves withthe samefrequency areproduced andradiate throughthe room. Aninterference

pattern is created(as representedin the diagram atthe right). The




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Lesson 1

Lesson 2

Lesson 3



ShockWaves


Lesson 4

Lesson 5


The Doppler Effect and Shock Waves

The Doppler effect is a phenomenonobserved whenever the source of waves ismoving with respect to an observer. TheDoppler effect can be described as theeffect produced by a moving source of

waves in which there is an apparent upwardshift in frequency for the observer and thesource are approaching and an apparentdownward shift in frequency when theobserver and the source is receding. TheDoppler effect can be observed to occurwith all types of waves - most notablywater waves, sound waves, and lightwaves. The application of this phenomenonto water waves was discussed in detail in

Unit 10 of The Physics Classroom; in thisunit, we will focus on the application of theDoppler effect to sound.

We are most familiar with the Dopplereffect because of our experiences withsound waves. Perhaps you recall aninstance in which a police car or emergencyvehicle was traveling towards you on thehighway. As the car approached with its

siren blasting, the pitch of the siren sound(a measure of the siren's frequency) washigh; and then suddenly after the carpassed by, the pitch of the siren sound waslow. That was the Doppler effect - anapparent shift in frequency for a soundwave produced by a moving source.







And perhaps you recall the laser discsegment shown in class involving theapproach of the snow plow with its blaringhorn. As the snow plow approach, thesound of its horn was hear at a high pitchand as the snow plow moved away, the

sound of its horn was heard at a low pitch.And finally, you might recall the nerf balldemonstration performed in class. A nerf ball was equipped with a buzzer whichproduced a sound with a constantfrequency. The ball was then througharound the room. As the ball approachedyou, you observed a higher pitch than whenthe ball was at rest. And when the ball wasthrown away from you, you observed a

lower pitch than when it was at rest. Theseare all examples of the Doppler effect. Butwhy does it happen?

The Doppler effect is observed because thedistance between the source of sound andthe observer is changing. If the source andthe observer are approaching, then thedistance is decreasing and if the source andthe observer are receding, then the

distance is increasing. The source of soundalways emits the same frequency.Therefore, for the same period of time, thesame number of waves must fit betweenthe source and the observer. if the distanceis large, then the waves can be spreadapart; but if the distance is small, thewaves must be compressed into the smallerdistance. For these reasons, if the source ismoving towards the observer, the observer

perceives sound waves reaching him or herat a more frequent rate (high pitch); and if the source is moving away from theobserver, the observer perceives sound




waves reaching him or her at a lessfrequent rate (low pitch). It is important tonote that the effect does not result becauseof an actual change in the frequency of the

source. The source puts out the samefrequency; the observer only perceives adifferent frequency because of the relativemotion between them.

The Doppler effect isobserved whenever thespeed of the source ismoving slower than thespeed of the waves. But if the source actually moves at

the same speed as or fasterthan the wave itself canmove, a differentphenomenon is observed. If a moving source of soundmoves at the same speed assound, then the source will always be at theleading edge of the waves which itproduces. The diagram at the right depictssnapshots in time of a variety of wavefronts

produced by an aircraft which is moving atthe same speed as sound. The circular linesrepresent compressional wavefronts of thesound waves. Notice that these circles arebunched up at the front of the aircraft. Thisphenomenon is known as a shock wave.Shock waves are also produced if theaircraft moves faster than the speed of sound. If a moving source of sound movesfaster than sound, the source will always be

ahead of the waves which it produces. Thediagram at the right depicts snapshots intime of a variety of wavefronts produced byan aircraft which is moving faster than




sound. Note that the circular compressionalwavefronts fall behind the faster movingaircraft (in actuality, these circles would bespheres).

If you are standing on the ground when asupersonic (faster than sound) aircraftpasses overhead, you might hear a sonicboom. A sonic boom occurs as the resultof the piling up of compressionalwavefronts along the conical edge of thewave pattern. These compressionalwavefronts pile up and interfere to producea very high pressure zone. This is shown

below. Instead of these compressionalregions (high pressure regions) reachingyou one at a time in consecutive fashion,they all reach you at once. Since everycompression is followed by a rarefaction,the high pressure zone will be immediatelyfollowed by a low pressure zone. Thiscreates a very loud noise.

If you are standing on the ground as thesupersonic aircraft passes by, there will bea short time delay and then you will hearthe boom - the sonic boom. This boom ismerely a loud noise resulting from the high

pressure sound followed by a low pressuresound. Do not be mistaken into thinkingthat this boom only happens the instantthat the aircraft surpasses the speed of sound and that it is the signature that theaircraft just attained supersonic speed.Sonic booms are observed when anyaircraft which is traveling faster than thespeed of sound passes overhead. It is not asign that the aircraft just overcame the

sound barrier, but rather a sign that theaircraft is traveling faster than sound.





Suppose a train is approaching you as you

stand on the loading platform at the railwaystation. As the train approaches, it slowsdown. All the while, the engineer issounding the horn at a constant frequencyof 500 Hz. Describe the pitch and thechanges in pitch that you hear.

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Lesson 1

Lesson 2

Lesson 3



ShockWaves


Lesson 4

Lesson 5


Boundary Behavior

As a sound wave travels through a medium,it will often reach the end of the mediumand encounter an obstacle or perhapsanother medium through which it couldtravel. When one medium ends, another

medium begins; the interface of the twomedia is referred to as the boundary andthe behavior of a wave at that boundary isdescribed as its boundary behavior. Thebehavior of a wave (or pulse) uponreaching the end of a medium is referred toas boundary behavior. There areessentially four possible boundarybehaviors by which a sound wave couldbehave: reflection (the bouncing off of the

boundary), diffraction (the bending aroundthe obstacle without crossing over theboundary), transmission (the crossing of the boundary into the new material orobstacle), and refraction (occurs along withtransmission and is characterized by thesubsequent change in speed and direction).In this part of Lesson 3, the focus will beupon the reflection behavior of soundwaves. Later in Lesson 3, diffraction,

transmission, and refraction will bediscussed in more detail.

In Unit 10 of The Physics Classroom, theboundary behavior of a pulse on a rope wasdiscussed. In thatunit, it wasmentioned that thereare two types of reflection for waves

on ropes: fixed end reflection and free endreflection. A pulse moving through a ropewill eventually reach its end. Upon reachingthe end of the medium, two things occur:




A portion of the energy carried by thepulse is reflected and returns towardsthe left end of the rope. The disturbance

which returns to the left is known as thereflected pulse.

A portion of the energy carried by thepulse is transmitted into the newmedium. If the rope is attached to apole (as shown at the right), the polewill receive some of the energy andbegin to vibrate. If the rope is notattached to a pole but rather resting onthe ground, then a portion of the energy

is transmitted into the air (the newmedium), causing slight disturbances of the the air particles.

In the situation in which the rope isattached to the pole, fixed end reflectionoccurs. For the reflected pulse off the fixedend, there is one very notable observation:the reflected pulse is inverted. That is, if acrest is incident towards a fixed end, it will

reflect and return as a trough. Similarly, if a trough is incident towards a fixed end, itwill reflect and return as a crest.

In the situation in which the rope is notattached to the pole nor firmly attached toany other medium heavier than itself (suchas another rope), free end reflectionoccurs. The reflected pulse off a free end is

never inverted. That is, if a crest is incidenttowards a free end, it will reflect and returnas a crest. Similarly, if a trough is incidenttowards a free end, it will reflect and returnas a trough.

In each case (free and fixed end reflection,the amount of energy which becomes

reflected is dependent upon thedissimilarity of the two medium. The moresimilar that the two medium on each side of the boundary are, the less reflection which




occurs and the more transmission whichoccurs. Conversely, the less similar that thetwo medium on each side of the boundaryare, the more reflection which occurs and

the less transmission which occurs. So if aheavy rope is attached to a light rope (twovery dissimilar medium), little transmissionand mostly reflection occurs. And if a heavyrope is attached to another heavy rope (twovery similar medium), little reflection andmostly transmission occurs.

The more similar the medium, the moretransmission which occurs.

These principles of free and fixed endreflection can be applied to sound waves.Though a sound wave does not consist of crests and troughs, they do consist of compressions and rarefactions. If a soundwave is traveling through a cylindrical tube,it will eventually come to the end of the

tube. The end of the tube represents aboundary between the enclosed air in thetube and the expanse of air outside of thetube. Upon reaching the end of the tube,the sound wave will undergo partialreflection and partial transmission. That is,a portion of the energy carried by thesound wave will pass across the boundaryand out of the tube (transmission) and aportion of the energy carried by the sound

wave will reflect off the boundary, remainin the tube and travel in the oppositedirection (reflection). If the end of the tubeis "open" or uncovered such that the air at




the end of the tube can freely vibrate whenthe sound wave reaches it, then thebehavior at the boundary resembles freeend reflection. There is no inversion of the

disturbance when reflecting off the openend (uncovered end) of a cylindrical tube.That is, if a compression is incident towardsan open-end, it will reflect and return as acompression. Similarly, if a rarefaction isincident towards an open end, it will reflectand return as a rarefaction. The oppositeoccurs if the end of the tube is "closed" orcovered up. If the end of the tube is"closed" or covered, then the air at the end

of the tube is fixed and cannot freelyvibrate when the sound wave reaches it. Inthis case, the behavior at the boundaryresembles fixed end reflection. There isinversion of the disturbance when reflectingoff the closed end (covered end) of acylindrical tube. That is, if a compression isincident towards an closed end, it willreflect and return as a rarefaction.Similarly, if a rarefaction is incident towards

an closed end, it will reflect and return as acompression.

The behavior of sound waves at open ends

and closed ends will become importantLesson 5 during the discussion of musicalinstruments. Many musical instrumentsoperate as the result of sound wavestraveling back and forth inside of "tubes" orair columns. These waves reflect at either aclosed end or an open end of the aircolumn; and the fact that inversion occursat a closed end will have a huge impact onthe numerical pattern of frequencies

produced by such instruments.




The reflection of sound also becomesimportant to the design of concert halls andauditoriums. The acoustics of sound must

be considered in the design of suchbuildings. The most importantconsiderations include destructiveinterference and reverberations, both of which are the result of reflections of soundoff the walls and ceilings. Designersattempt to reduce the severity of theseproblems by using building materials whichreduce the amount of reflection andenhance the amount of transmission (or

absorption) into the walls and ceilings. Themost reflective materials are those whichare smooth and hard; such materials arevery dissimilar to air and thus reduce theamount of transmission and increase theamount of reflection. The best materials touse in the design of concert halls andauditoriums are those materials which aresoft. For this reason, fiberglass and acoustictile are used in such buildings rather than

cement and brick.







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Lesson 4

Lesson 5


Reflection, Refraction, and Diffraction

Like any wave, a sound wave doesn't juststop when it reaches the end of themedium or when it encounters an obstaclein its path. Rather, a sound wave willundergo certain behaviors when it

encounters the end of the medium or anobstacle. Possible behaviors includereflection off the obstacle, diffractionaround the obstacle, and transmission(accompanied by refraction) into theobstacle or new medium . In this part of Lesson 3, we will investigate behaviorswhich have already been discussed in aprevious unit and apply them towards thereflection, diffraction, and refraction of

sound waves.

When a wave reaches the boundarybetween one medium another medium, aportion of the wave undergoes reflectionand a portion of the wave undergoestransmission across the boundary. Asdiscussed in the previous part of Lesson 3,the reflected wave may or may not undergoa phase change (i.e., be inverted)

depending on the relative densities of thetwo media. It was also mentioned that theamount of reflection is dependent upon thedissimilarity of the two medium. For thisreason, acousticly minded builders of auditoriums and concert halls avoid the useof hard, smooth materials in theconstruction of their inside halls. A hardmaterial such as concrete is as dissimilar ascan be to the air through which the sound

moves; subsequently, most of the soundwave is reflected by the walls and little isabsorbed.




Reflection of sound waves off of surfacescan lead to one of two phenomenon - anecho or a reverberation. A reverberation

often occurs in a small room with height,width, and length dimensions of approximately 17 meters or less. Why themagical 17 meters? The effect of aparticular sound wave upon the brainendures for more than a tiny fraction of asecond; the human brain keeps a sound inmemory for up to 0.1 seconds. If a reflectedsound wave reaches the ear within 0.1seconds of the initial sound, then it seems

to the person that the sound is prolonged .The reception of multiple reflections off of walls and ceilings within 0.1 seconds of each other causes reverberations - theprolonging of a sound. Since sound wavestravel at about 340 m/s at roomtemperature, it will take approximately 0.1s for a sound to travel the length of a 17meter room and back, thus causing areverberation (recall from Lesson 2, t = v/d

= (340 m/s)/(34 m) = 0.1 s). This is whyreverberations is common in rooms withdimensions of approximately 17 meters orless. Perhaps you have observedreverberations when talking in an emptyroom, when honking the horn while drivingthrough a highway tunnel or underpass, orwhen singing in the shower. In auditoriumsand concert halls, reverberationsoccasionally occur and lead to the

displeasing garbling of a sound.

But reflection of sound waves inauditoriums and concert halls do not alwayslead to displeasing results, especially if thereflections are designed right . Smooth wallshave a tendency to direct sound waves in aspecific direction. Subsequently the use of smooth walls in an auditorium will causespectators to receive a large amount of

sound from one location along the wall;there would be only one possible path bywhich sound waves could travel from thespeakers to the listener. The auditorium




would not seem to be as lively and full of sound. Rough walls tend to diffuse sound,reflecting it in a variety of directions. Thisallows a spectator to perceive sounds from

every part of the room, making it seemlively and full. For this reason, auditoriumand concert hall designers preferconstruction materials which are roughrather than smooth.

Reflection of sound waves also lead toechoes. Echoes are different thanreverberations. Echoes occur when areflected sound wave reaches the ear morethan 0.1 seconds after the original soundwave was heard. If the elapsed timebetween the arrival of the two sound wavesis more than 0.1 seconds, then thesensation of the first sound will have died out . In this case, the arrival of the secondsound wave will be perceived as a secondsound rather than the prolonging of the firstsound. There will be an echo instead of areverberation.

Reflection of sound waves off of surfaces is

also effected by the shape of the surface.As mentioned of water waves in Unit 10,flat or plane surfaces reflect sound waves insuch a way that the angle at which thewave approaches the surface equals theangle at which the wave leaves the surface.This principle will be extended to thereflective behavior of light waves off of plane surfaces in great detail in Unit 13 of The Physics Classroom. Reflection of sound

waves off of curved surfaces leads to amore interesting phenomenon. Curvedsurfaces with a parabolic shape have thehabit of focusing sound waves to a point.




Sound waves reflecting off of parabolicsurfaces concentrate all their energy to asingle point in space; at that point, thesound is amplified. Perhaps you have seen

a museum exhibit which utilizes aparabolic-shaped disk to collect a largeamount of sound and focus it at a focalpoint. If you place your ear at the focalpoint, you can hear even the faintestwhisper of a friend standing across theroom. Parabolic-shaped satellite disks usethis same principle of reflection to gatherlarge amounts of electromagnetic wavesand focus it at a point (where the receptor

is located). Scientists have recentlydiscovered some evidence which seem toreveal that the bull moose utilizes hisantlers as a satellite disk to gather andfocus sound. Finally, scientists have longbelieved that owls are equipped withspherically-shaped facial disks which can bemaneuvered in order to gather and reflectsound towards their ears. This principle willbe extended to the reflective behavior of

light waves off curved surfaces in greatdetail in Unit 13 of The Physics Classroom.

Diffraction involves a change in directionof waves as they pass through an openingor around a barrier in their path. Thediffraction of water waves was discussed inUnit 10 of The Physics Classroom. In thatunit, we saw that water waves have theability to travel around corners, around

obstacles and through openings. Theamount of diffraction (the sharpness of thebending) increases with increasingwavelength and decreases with decreasingwavelength. In fact, when the wavelengthof the waves are smaller than the obstacleor opening, no noticeable diffraction occurs.

Diffraction of sound waves iscommonly observed; we notice

sound diffracting around cornersor through door openings,allowing us to hear others whoare speaking to us from




adjacent rooms. Many forest-dwelling birdstake advantage of the diffractive ability of long-wavelength sound waves. Owls forinstance are able to communicate across

long distances due to the fact that theirlong-wavelength hoots are able to diffractaround forest trees and carry farther thanthe short-wavelength tweets of song birds.Low-pitched (high wavelength) soundsalways carry further than high pitched (lowwavelength) sounds.

Scientists have recently learned thatelephants emit infrasonic waves of very low

frequency to communicate over longdistances to each other. Elephants typicallymigrate in large herds which maysometimes become separated from eachother by distances of several miles.Researchers who have observed elephantmigrations from the air have beenimpressed and puzzled by the ability of elephants at the beginning and the end of these herds to make extremely

synchronized movements. The matriarch atthe front of the heard might make a turn tothe right which is immediately followed byelephants at the end of the herd making thesame turn to the right. These synchronizedmovements occur despite the fact that theelephants' vision of each other is blocked bydense vegetation. Only recently have theylearned that the synchronized movementsare preceded by infrasonic communication.

While low wavelength light waves areunable to diffract around the densevegetation, the high wavelength soundsproduced by the elephants have sufficientdiffractive ability to communicate longdistances.

Bats use high frequency (low wavelength)ultrasonic waves in order to enhance theirability to hunt. The typical prey of a bat is

the moth - an object not much larger than acouple of centimeters. Bats use ultrasonicecholocation methods to detect thepresence of bats in the air. But why




ultrasound? The answer lies in the physicsof diffraction. As the wavelength of a wavebecomes smaller than the obstacle which itencounters, the wave is no longer able to

diffract around the obstacle, instead thewave reflects off the obstacle. Bats useultrasonic waves with wavelengths smallerthan the dimensions of their prey. Thesesound waves will encounter the prey, andinstead of diffracting around the prey, willreflect off the prey and allow the bat tohunt by means of echolocation. Thewavelength of a 50 000 Hz sound wave inair (speed of approximately 340 m/s) can

be calculated as follows

wavelength = speed/frequency

wavelength = (340 m/s)/(50 000 Hz)

wavelength = 0.0068 m

The wavelength of the 50 000 Hz soundwave (typical for a bat) is approximately0.7 centimeters, smaller than the

dimensions of a typical moth.

Refraction of waves involves a change inthe direction of waves as they pass fromone medium to another. Refraction, orbending of the path of the waves, isaccompanied by a change in speed andwavelength of the waves. So if the medium(and its properties) are changed, the speedof the waves are changed. Thus wavespassing from one medium to another willundergo refraction. Refraction of soundwaves is most evident in situations in whichthe sound wave passes through a mediumwith gradually varying properties. Forexample, sound waves are known to refractwhen traveling over water. Even though thesound wave is not exactly changing media,

it is traveling through a medium withvarying properties; thus,the wave will encounterrefraction and change its




direction. Since water has a moderatingeffect upon the temperature of air, the airdirectly above the water tends to be coolerthan the air far above the water. Sound

waves travel slower in cooler air than theydo in warmer air. For this reason, theportion of the wavefront directly above thewater is slowed down, while the portion of the wavefronts far above the water speedsahead. Subsequently, the direction of thewave changes, refracting downwardstowards the water. This is depicted in thediagram at the right.

Refraction of other waves such as lightwaves will be discussed in more detail in alater Unit 14 of The Physics Classroom.

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NaturalFrequency

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VibrationStandingWavePatternsFundamentalFrequencyandHarmonics

Lesson 5

Lesson 4: Resonance and Standing

Waves

Natural Frequency

As has been previously mentioned in thisunit, a sound wave is created as a result of a vibrating object. The vibrating object isthe source of the disturbance which moves

through the medium. The vibrating objectwhich creates the disturbance could be thevocal chords of a person, the vibratingstring and sound board of a guitar or violin,the vibrating tines of a tuning fork, or thevibrating diaphragm of a radio speaker. Anyobject which vibrates will create a sound.The sound could be musical or it could benoisy; but regardless of its quality, thesound was created by a vibrating object.

Nearly all objects, when hit or struck orplucked or strummed or somehowdisturbed, will vibrate. If you drop a meterstick or pencil on the floor, it will begin tovibrate. If you pluck a guitar string, it willbegin to vibrate. If you blow over the top of a pop bottle, the air inside will vibrate.When each of these objects vibrate, theytend to vibrate at a particular frequency or

a set of frequencies. The frequency orfrequencies at which an object tends tovibrate with when hit, struck, plucked,strummed or somehow disturbed is knownas the natural frequency of the object. If the amplitude of the vibrations are largeenough and if natural frequency is withinthe human frequency range, then theobject will produce sound waves which areaudible.

All objects have a natural frequency or setof frequencies at which they vibrate. Thequality or timbre of the sound produced by




a vibrating object is dependent upon thenatural frequencies of the sound wavesproduced by the objects. Some objects tendto vibrate at a single frequency and they

are often said to produce a pure tone. Aflute tends to vibrate at a single frequency,producing a very pure tone. Other objectsvibrate and produce more complex waveswith a set of frequencies which have awhole number mathematical relationshipbetween them; these are said to produce arich sound. A tuba tends to vibrate at a setof frequencies which are mathematicallyrelated by whole number ratios; it produces

a rich tone. Still other objects will vibrate ata set of multiple frequencies which have nosimple mathematical relationship betweenthem. These objects are not musical at alland the sounds which they create are bestdescribed as noise. When a meter stick orpencil is dropped on the floor, a vibrateswith a number of frequencies, producing acomplex sound wave which is clanky andnoisy.

The actual frequency at which an object willvibrate at is determined by a variety of factors. Each of these factors will eithereffect the wavelength or the speed of theobject. Since

frequency = speed/wavelength

an alteration in either speed or wavelengthwill result in an alteration of the naturalfrequency. The role of a musician is tocontrol these variables in order to producea given frequency from the instrument




which is being played. Consider a guitar asan example. There are six strings, eachhaving a different linear density (the widerstrings are more dense on a per meter

basis), a different tension (which iscontrollable by the guitarist, and a differentlength (also controllable by the guitarist).The speed at which waves move throughthe strings is dependent upon theproperties of the medium - in this case thetightness (tension) of the string and thelinear density of the strings. Changes inthese properties would effect the naturalfrequency of a particular string. The

vibrating portion of a particular string canbe shortened by pressing the string againstone of the frets on the neck of the guitar;this modification in the length of the stringwould effect the wavelength of the waveand in turn the natural frequency at which aparticular string vibrates at. Controlling thespeed and the wavelength in this mannerallows a guitarist to control the naturalfrequencies of the vibrating object (a

string) and thus produce the intendedmusical sounds. The same principles can beapplied to any string instrument - whetherit be the piano, harp, harpsichord, violin orguitar.

As another example, consider the trombonewith its long cylindrical tube which is bentupon itself twice and ends in a flared end.The trombone is an example of a wind

instrument. The "tube" of any windinstrument acts as a container for avibrating air column; the air inside the tubewill be set into vibrations by a vibratingreed or the vibrations of a musicians lipsagainst a mouthpiece. While the speed of sound waves within the air column is notalterable by the musician (they can only bealtered by changes in room temperature),the length of the air column is. For a

trombone, the length is altered by pushingthe tube outward away from themouthpiece to lengthen it or pulling it in toshorten it. This causes the length of the air




column to be changed, and subsequentlychanges the wavelength of the waves itproduces. And of course, a change inwavelength will result in a change in the

frequency. So the natural frequency of awind instrument such as the trombone isdependent upon the length of the aircolumn of the instrument. The sameprinciples can be applied to any windinstrument -whether it be the tuba, flute,wind chime, organ pipe, clarinet, or popbottle.

There were a variety of classroom demonstrations(some of which were phunand some of which werecorny) which illustrated the

idea of natural frequencies and theirmodification. First recall the pop bottleinstrument. A pop bottle was partly filledwith water leaving a column of air insidewhich was capable of vibrating. When airwas blown over the top of the instrument,the air inside was set into vibrationalmotion (turbulence above the lip of thebottle creates disturbances within thebottle). These vibrations resulted in a

sound wave which was audible to students.Of course, the frequency can be modifiedby altering the length of the air column(adding or removing water) which changesthe wavelength and in turn the frequency.As we know from our understanding of thefrequency-wavelength relation, a shorter aircolumn means a shorter wavelength and ahigher frequency.

Then there was the toilet paper roll medley.Different lengths of toilet paper rolls (orwrapping paper rolls) will vibrate withdifferent frequencies when struck against a




students head. A properly selected set of rolls will result in the production of soundswhich are capable of a Tony Awardrendition of "Mary Had a Little Lamb."

Maybe you are familiar with the popularwater goblet prom trick which wasdemonstrated in class. Obtain awater goblet and clean yourfingers. Then gently slide yourfinger over the rim of the watergoblet. If you are fortunateenough, you might be able toset the goblet into vibration by means of

slip-stick friction. (It is not necessary touse a crystal goblet; it is often said thatcrystal goblets work better, but I have beenable to perform the trick just as easily withclean fingers and an inexpensive goblet.)Like a violin bow string being pulled acrossa violin string, the finger sticks to the glassmolecules, pulling them apart at a givenpoint until the tension becomes so great.The finger then slips off the glass and

subsequently finds another microscopicsurface to stick to; the finger pulls themolecules at that surface, slips and thensticks at another location. This process of stick-slip friction is sufficient to set themolecules in the glass into vibration at itsnatural frequency. The result is enough toimpress your dinner guests. Try it athome!!

Perhaps you recall a simple pendulumdemonstration. While a pendulum does notproduce a noise when it oscillates, it doesillustrate an important principle. Thependulums consisting of the longer stringsvibrate with a longer period and thus alower frequency. Once more, there is aninverse relationship between the length of the vibrating object and the naturalfrequency at which the object vibrates. This

very relationship carries over to anyvibrating instrument - whether it be aguitar string, a xylophone, a pop bottleinstrument, or a kettle drum.




To conclude, all objects have a naturalfrequency or set of frequencies at which

they vibrate when struck, plucked,strummed or somehow disturbed. Theactual frequency is dependent upon theproperties of the material the object ismade of (this effects the speed of thewave) and the length of the material (thiseffects the wavelength of the wave). It isthe goal of musicians to find instrumentswhich possess the ability to vibrate withsets of frequencies which are musicallysounding (i.e., mathematically related bysimple whole number ratios) and to varythe lengths and (if possible) properties tocreate the desired sounds.







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NaturalFrequency

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Waves

Forced Vibration

Musical instruments and other objects areset into vibration at their natural frequencywhen a person hits, strikes, strums, plucksor somehow disturbs the object. For

instance, a guitar string is strummed orplucked; a piano string is hit with ahammer when a pedal is played; and thetines of a tuning fork are hit with a rubbermallet. Whatever the case, a person orthing puts energy into the instrument bydirect contact with it. This input of energydisturbs the particles and forces the objectinto vibrational motion - at its naturalfrequency.

If you were to take a guitar string andstretch it to a given length and a giventightness and have a friend pluck it, youwould hear a noise; but the noise would noteven be close in comparison to theloudness produced by an acoustic guitar.On the other hand, if the string is attachedto the sound box of the guitar, the vibratingstring is capable of forcing the sound box

into vibrating at that same naturalfrequency. The sound box in turn forces airparticles inside the box into vibrationalmotion at the same natural frequency asthe string. The entire system (string, guitar,and enclosed air) begins vibrating andforces surrounding air particles intovibrational motion. The tendency of oneobject to force another adjoining orinterconnected object into vibrational

motion is referred to as a forcedvibration. In the case of the guitar stringmounted to the sound box, the fact that thesurface area of the sound box is greater




than the surface area of the string, meansthat more surrounding air particles will beforced into vibration. This causes anincrease in the amplitude and thus loudness

of the sound.

This same principle of aforced vibration wasdemonstrated in class usinga tuning fork. If the tuningfork is held in your handand hit with a hammer, asound is produced as thetines of the tuning fork set

surrounding air particlesinto vibrational motion. The sound producedby the tuning fork is barely audible tostudents in the back rows of the room.However, if the tuning fork is set upon thewhiteboard panel or the glass panel of theoverhead projector, the panels beginvibrating at the same natural frequency of the tuning fork. The tuning fork forcessurrounding glass (or vinyl) particles into

vibrational motion. The vibratingwhiteboard or overhead projector panel inturn forces surrounding air particles intovibrational motion and the result is anincrease in the amplitude and thus loudnessof the sound. This principle of forcedvibration explains why the classroom tuningfork is mounted on a sound box, why acommercial music box mechanism ismounted on a sounding board, why a guitar

utilizes a sound box, and why a piano stringis attached to a sounding board - a loudersound is always produced.

Now consider a related situation whichresembles another classroomdemonstration. Suppose that a tuning forkis mounted on a sound box and set uponthe table; and suppose a second tuningfork/sound box system having the same

natural frequency (say 256 Hz) is placed onthe table near the first system. Neither of the tuning forks is vibrating. Then the firsttuning fork is struck with a rubber mallet




and the tines begin vibrating at its naturalfrequency - 256 Hz. These vibrations setthe sound box and the air inside the soundbox vibrating at the

same naturalfrequency of 256 Hz.Surrounding airparticles are set intovibrational motion atthe same naturalfrequency of 256 Hz and every student inthe classroom hears the sound. Then thetines of the tuning fork are grabbed toprevent their vibration and remarkably the

sound of 256 Hz is still being heard. Onlynow the sound is being produced by thesecond tuning fork - the one which wasn'thit with the mallet. Amazing!! In fact, it isso amazing, that the demonstration isrepeated to assure that the same surprisingresults are observed. They are! What ishappening?

In this demonstration, one tuning fork

forces another tuning fork into vibrationalmotion at the same natural frequency. Thetwo forks are connected by the surroundingair particles. As the air particlessurrounding the first fork (and itsconnected sound box) begin vibrating, thepressure waves which it creates begin toimpinge at a periodic and regular rate of 256 Hz upon the second tuning fork (and itsconnected sound box). The energy carried

by this sound wave through the air is tuned to the frequency of the second tuning fork.Since the incoming sound waves share thesame natural frequency as the secondtuning fork, the tuning fork easily beginsvibrating at its natural frequency. This is anexample of resonance - when one objectvibrating at the same natural frequency of asecond object forces that second object intovibrational motion.

The result of resonance is always a largevibration. Regardless of the vibratingsystem, if resonance occurs, a large




vibration results. This was demonstrated inclass with an odd-looking mechanicalsystem resembling an inverted pendulum.Three sets of two identical plastic bobs

were mounted on a veryelastic metal pole, whichwere in turn mounted toa metal bar. Each metalpole and attached bobhad a different length,thus giving it a differentnatural frequency of vibration. The bobswere color coded to distinguish betweenthem - they were colored red, blue and

green (this will be significant later in thecourse). The red bobs were mounted on thelonger poles and they had the lowestnatural frequency of vibration. The bluebobs were mounted on the shorter polesand they had the highest natural frequencyof vibration. (Note the length-wavelength-frequency relationship that was discussedearlier.) When the red bob was disturbed, itbegan vibrating at its natural frequency,

which in turn caused the attached bar tovibrate at the same frequency; this in turnset the other attached red bob intovibrating at the same natural frequency.This is resonance - one bob vibrating at agiven frequency forcing a second objectwith the same natural frequency intovibrational motion. While the green and theblue bobs were disturbed by the vibrationstransmitted through the metal bar, only the

red bob would resonate. This is because thefrequency of the first red bob was tuned tothe frequency of the second red bob; theyshare the same natural frequency. Theresult was that the second red bob beginsvibrating with a huge amplitude.







Another classroomdemonstration of resonanceinvolved a plastic tube which

was raised an lowered in acylinder of water until thevibrations of the air inside of the tube was tuned to thevibrations of a tuning fork. Byraising and lowering theplastic tube, the naturalfrequency at which air insideof the tube could vibrate wasmodified. If the tube is raised, thus

increasing the length of the air columninside of the tube, the natural frequency of the air column is decreased. Conversely, if the tube is lowered into the water, thusdecreasing the length of the air columninside of the tube, the natural frequency of the air column is increased. (Again note thelength-wavelength-frequency relationshipthat was discussed earlier.) While theraising and lowering of the tube into and

out of the cylinder is being carried on, avibrating tuning fork is held above the aircolumn. When the natural frequency of theair column is tuned to the frequency of thevibrating tuning fork, resonance occurs anda loud sound results. The vibrating tuningfork forces air particles within the aircolumn into vibrational motion. Once morein this resonance situation, the tuning forkand the air column share the same

vibrational frequency.

In conclusion, resonance occurs when twointerconnected objects share the samevibrational frequency. When one of theobjects is vibrating, it forces the secondobject into vibrational motion. The result isa large vibration, and if a sound wavewithin the audible range of human hearingis produced, a loud sound is heard.




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NaturalFrequency

Forced


Lesson 5


Waves

Standing Wave Patterns

As mentioned earlier, all objects have afrequency or set of frequencies with whichthey naturally vibrate when struck, plucked,strummed or somehow disturbed. Each of

the natural frequencies at which an objectvibrates is associated with a standing wavepattern. When an object is forced intoresonance vibrations at one of its naturalfrequencies, it vibrates in a manner suchthat a standing wave is formed within theobject. The topic of standing wave patternswas introduced in Unit 10 of The PhysicsClassroom. In that unit, a standing wavepattern was described as a vibrational

pattern created within a medium when thevibrational frequency of a source causesreflected waves from one end of themedium to interfere with incident wavesfrom the source in such a manner thatspecific points along the medium appear tobe standing still. Such patterns are onlycreated within the medium at specificfrequencies of vibration; these frequenciesare known as harmonic frequencies, or

merely harmonics. At any frequency otherthan a harmonic frequency, theinterference of reflected and incident wavesresults in a resulting disturbance of themedium which is irregular and non-repeating.

So the natural frequencies of an object aremerely the harmonic frequencies at whichstanding wave patterns are established

within the object. These standing wavepatterns represent the lowest energyvibrational modes of the object. Whilethere are countless way by which an object




can vibrate (each associated with a specificfrequency), objects favor only a few specificmodes or patterns of vibrating. The favoredmodes (patterns) of vibration are those

which result in the highest amplitudevibrations with the least input of energy.Objects favor these natural modes of vibration because they are representativeof the patterns which require the leastamount of energy. Objects are most easilyforced into resonance vibrations whendisturbed at frequencies associated withthese natural frequencies.

The wave pattern associated with thenatural frequencies of an object ischaracterized by points which appear to bestanding still; for this reason, the pattern isoften called a "standing wave pattern." Thepoints in the pattern which are standingstill are referred to as nodal points or nodalpositions. These positions occur as theresult of the destructive interference of incident and reflected waves. Each nodal

point is surrounded by anti-nodal points,creating an alternating pattern of nodal andanti-nodal points. Such patterns wereintroduced in Unit 10 of The PhysicsClassroom. In this unit, we will elaborate onthe essential characteristics and the causesof standing wave patterns and relate thesepatterns to the vibrations of musicalinstruments.

Perhaps you recall the classroomdemonstration utilizing the square metalplate (known as a Chladni plate), a violinbow and the salt. The plate was securelyfastened to a table using a nut and bolt; thenut and bolt were clamped to thecenter of the square plate,preventing that section fromvibrating. The salt was sprinkledupon the plate in an irregular

pattern. Then the violin bow wasused to induce vibrations within theplate; the plate was strummed andbegan vibrating. And then the magic




occurred. A high-pitched pure tone wassounded out as the plate vibrated; and,remarkably (as is often the case in physicsclass), the salt upon the plate began

vibrating and formed a pattern upon theplate. As we know, all objects (even a sillylittle metal plate) have a set of naturalfrequencies at which they vibrate; and eachfrequency is associated with a standingwave pattern. The pattern formed by thesalt on the plate was the standing wavepattern associated with one of the naturalfrequencies of the Chladni plate. As theplate began to vibrate, the salt began to

vibrate and tumble about the plate untilthey reached points along the plate whichwere not vibrating. Subsequently, the saltfinally comes to rest along the nodalpositions. The diagrams at the right showtwo of the most common standing wavepatterns for the Chladni plates. The whitelines represent the salt locations (nodalpositions). Observe in the diagram thateach pattern is characterized by nodal

positions in the corners of the square plateand in the center of the plate. For thesetwo particular vibrational modes, thosepositions are unable to move. Being unableto move, they become nodal points - pointsof no displacement.

The diagram below depicts one of thenatural patterns of vibrations for a guitarstring. In the pattern, you will note thatthere are certain positions along the string(the medium) in which points appear to bestanding still. These points are referred toas nodal points and are labeled on thediagram. In between each nodal position,are other positions which appear to be

vibrating back and forth between a largeupward displacement to a large downwarddisplacement. These points are referred toas anti-nodes and are also labeled on the




diagram. There is an alternating pattern of nodal and anti-nodal positions in a standingwave pattern.

Because the anti-nodal positions along theguitar string are vibrating back and forthbetween a a large upward displacement toa large downward displacement, thestanding wave pattern is often depicted bya diagram such as that shown below.

The pattern above is not the only pattern of vibration for a guitar string. There are avariety of patterns by which

the guitar string couldnaturally vibrate; each patternis associated with one of thenatural frequencies of theguitar strings. Three otherpatterns are shown in thediagrams at the right. Eachstanding wave pattern is referred to as aharmonic of the instrument (in this case,the guitar string). The three diagrams at

the right represent the standing wavepatterns for the first, second, and thirdharmonics of a guitar string. (Harmonicswill be discussed in more detail in the nextsection of this lesson.) There are a varietyof other low energy vibrational patternswhich could be established in the string; forguitar strings, each pattern is characterizedby some basic traits:

There is an alternating patterns of nodes and antinodes.

There are either a half-number or awhole-number of waves within the




pattern established on the string.

Nodal positions (points of nodisplacement) are established at theends of the string where the string is

clamped down in a fixed position.One pattern is related to the nextpattern by the addition (or subtraction)of one or more nodes (and anti-nodes).

The standing wave patterns for othermusical instruments share some thesesame traits or at least similar traits. These

patterns will be discussed in more detail inLesson 5 of this unit.

1stHarmonic

2ndHarmonic

3rdHarmonic







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Lesson 1

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Lesson 4

NaturalFrequency

Forced


Lesson 5


Waves

Fundamental Frequency and Harmonics

Previously in Lesson 4, it was mentionedthat when an object is forced intoresonance vibrations at one of its naturalfrequencies, it vibrates in a manner such

that a standing wave pattern is formedwithin the object. Whether it be a guitarsting, a Chladni plate, or the air columnenclosed within a trombone, the vibratingmedium vibrates in such a way that astanding wave pattern results. Each naturalfrequency which an object or instrumentproduces has its own characteristicvibrational mode or standing wave pattern.These patterns are only created within the

object or instrument at specific frequenciesof vibration; these frequencies are knownas harmonic frequencies, or merelyharmonics. At any frequency other than aharmonic frequency, the resultingdisturbance of the medium is irregular andnon-repeating. For musical instruments andother objects which vibrate in regular andperiodic fashion, the harmonic frequenciesare related to each other by simple whole

number ratios. This is part of the reasonwhy such instruments sound musical ratherthan noisy. We will see in this part of Lesson 4 why these whole number ratiosexist for a musical instrument.

First, consider a guitar string vibrating at itsnatural frequency or harmonic frequency.Because the ends of the string are attachedand fixed in place to the guitar's structure

(the bridge at one end and the frets at theother), the ends of the string are unable tomove. Subsequently, these ends becomenodes - points of no displacement. In




between these two nodes at the end of thestring, there must be at least one anti-node. The most fundamental harmonic for aguitar string is the harmonic associated

with a standing wave having only one anti-node positioned between the two nodes onthe end of the string. Thiswould be the harmonic withthe longest wavelength andthe lowest frequency. Thelowest frequency producedby any particular instrument is known asthe fundamental frequency. Thefundamental frequency is alternatively

called the first harmonic of theinstrument. The diagram at the right showsthe first harmonic of a guitar string. If youanalyze the wave pattern in the guitarstring for this harmonic, you will notice thatthere is not quite one complete wave withinthe pattern. A complete wave starts at therest position, rises to a crest, returns torest, drops to a trough, and finally returnsto the rest position before starting its next

cycle. (Caution: the use of the words crestand trough to describe the pattern are onlyused to help identify the length of arepeating wave cycle. A standing wavepattern is not actually a wave, but rather apattern of a wave Thus, it does not consistsof crests and troughs, but rather nodes andanti-nodes. The pattern is the result of theinterference of two waves to produce thesenodes and anti-nodes.) In this pattern,

there is only one-half of a wave within thelength of the string. This is the case for thefirst harmonic or fundamental frequency of a guitar string. The diagram below depictsthis length-wavelength relationship for thefundamental frequency of a guitar string.





The second harmonic of aguitar string is produced byadding one more node

between the ends of theguitar string. And of course, if a node isadded to the pattern, an anti-node must beadded as well in order to maintain analternating pattern of nodes and anti-nodes.In order to create a regular and repeatingpattern, that node must be located exactlymidway between the ends of the guitarstring. This additional node gives thesecond harmonic a total of three nodes and

two anti-nodes. The standing wave patternfor the second harmonic is shown at theright. A careful investigation of the patternreveals that there is exactly one full wavewithin the length of the guitar string. Forthis reason, the length of the string is equalto the length of the wave.

The third harmonic of a guitarstring is produced by addingtwo nodes between the endsof the guitar string. And of course, if two nodes are added to thepattern, two anti-nodes must be added aswell in order to maintain an alternatingpattern of nodes and anti-nodes. In order tocreate a regular and repeating pattern forthis harmonic, the two additional nodesmust be evenly spaced between the ends of the guitar string; this places them at the

one-third mark and the two-thirds markalong the string. These additional nodesgive the third harmonic a total of fournodes and three anti-nodes. The standingwave pattern for the third harmonic isshown at the right. A careful investigationof the pattern reveals that there is morethan one full wave within the length of theguitar string. In fact, there are three-halvesof a wave within the length of the guitar

string. For this reason, the length of thestring is equal to three-halves the length of the wave. The diagram below depicts this

2001 The Physics Classroom and Mathsoft Education and




length-wavelength relationship for thefundamental frequency of a guitar string.

After a discussion of the first threeharmonics, a pattern can be recognized.Each harmonic results in an additional nodeand antinode, and an additional half of a

wave within the string. If the number of waves in a string is known, then anequation relating the wavelength of thestanding wave pattern to the length of thestring can be algebraically derived.

This information is summarized in the tablebelow.

Harm.#

# of Waves

inString

# of Nodes

# of Anti-nodes

Length-WavelengthRelationship

1 1/2 2 1 Wavelength =(2/1)*L

21 or 2/

23 2

Wavelength =(2/2)*L

3 3/2 4 3Wavelength =

(2/3)*L

42 or 4/

25 4

Wavelength =(2/4)*L

5 5/2 6 5Wavelength =

(2/5)*L




The above discussion develops themathematical relationship between thelength of a guitar string and the wavelength

of the standing wave patterns for thevarious harmonics which could beestablished within the string. Now theselength-wavelength relationships will beused to develop relationships for the ratioof the wavelengths and the ratio of thefrequencies for the various harmonicsplayed by an string instrument (such as aguitar string).

Consider a 80-cm long guitar string whichhas a fundamental frequency (1stharmonic) of 400 Hz. For the first harmonic,the wavelength of the wave pattern wouldbe two times the length of the string (seetable above); thus, the wavelength is 160cm or 1.60 m. The speed of the standingwave can now be determined from thewavelength and the frequency. The speedof the standing wave is

speed = frequency * wavelength

speed = 400 Hz * 1.6 m

speed = 640 m/s

This speed of 640 m/s corresponds to thespeed of any wave within the guitar string.Since the speed of any wave is dependentupon the properties of the medium (and notupon the properties of the wave), everywave will have the same speed in thisstring regardless of its frequency and itswavelength. So the standing wave patternassociated with the second harmonic, thirdharmonic, fourth harmonic, etc. will alsohave this speed of 640 m/s. A change infrequency or wavelength will NOT cause achange in speed.

Using the table above, the wavelength of

the second harmonic (denoted by thesymbol W2) would be 0.8 m (the same as

the length of the string). The speed of theEngineering, Inc.




standing wave pattern (denoted by thesymbol v) is still 640 m/s. Now the waveequation can be used to determine thefrequency of the second harmonic (denoted

by the symbol f 2).



f 2 = v / W2

f 2 = (640 m/s)/(0.8 m)

f 2 = 800 Hz

This same process can be repeated for thethird harmonic. Using the table above, thewavelength of the third harmonic (denotedby the symbol W3) would be 0.533 m (two-

thirds of the length of the string). Thespeed of the standing wave pattern

(denoted by the symbol v) is still 640 m/s.Now the wave equation can be used todetermine the frequency of the thirdharmonic (denoted by the symbol f 3).



f 3 = v / W3

f 3 = (640 m/s)/(0.533 m)

f 3 = 1200 Hz

Now if you have been following along, youwill have recognized a pattern. Thefrequency of the second harmonic is two

times the frequency of the first harmonic.The frequency of the third harmonic is threetimes the frequency of the first harmonic.The frequency of the nth harmonic (where





n represents the harmonic # of any of theharmonics) is n times the frequency of thefirst harmonic. In equation form, this can bewritten as

f n = n * f 1

The inverse of this pattern exists for thewavelength values of the variousharmonics. The wavelength of the secondharmonic is one-half (1/2) the wavelengthof the first harmonic. The wavelength of thethird harmonic is on-third (1/3) thewavelength of the first harmonic. And the

wavelength of the nth harmonic is one-nth(1/n) the wavelength of the first harmonic.In equation form, this can be written as

Wn = (1/n) * W1

These relationships between wavelengthsand frequencies of the various harmonicsfor a guitar string are summarized in thetable below.

Harm.#

Freq.(Hz)

Wavelength(m)

Speed(m/s)

f n/f 1

Wn/

W1

1 400 1.60 640 1 1/1

2 800 0.800 640 2 1/2

3 1200 0.533 640 3 1/3

4 1600 0.400 640 4 1/4

5 2000 0.320 640 5 1/5

nn *400

(2/n)*(0.800)

640 n 1/n

The table above demonstrates that theindividual frequencies in the set of natural

frequencies produced by a guitar string arerelated to each other by whole numberratios. For instance, the first and secondharmonics have a 2:1 frequency ratio; the




second and the third harmonics have a 3:2frequency ratio; the third and the fourthharmonics have a 4:3 frequency ratio; andthe fifth and the fourth harmonic have a 5:4

frequency ratio. When the guitar is played,the string, sound box and surrounding airvibrate at a set of frequencies to produce awave with a mixture of harmonics. Theexact composition of that mixturedetermines the timbre or quality of soundwhich is heard. If there is only a singleharmonic sounding out in the mixture (inwhich case, it wouldn't be a mixture), thenthe sound is rather pure-sounding. On the

other hand, if there are a variety of frequencies sounding out in the mixture,then the timbre of the sound is rather richin quality.

In Lesson 5, these same principles of resonance and standing waves will beapplied to other types of instrumentsbesides guitar strings.


1. The sound produced byblowing over the top of apartially filled soda popbottle is the result of theair column inside of the bottle vibrating atits natural frequency. The actual frequency

of vibration is inversely proportional to thewavelength of the sound; and thus, thefrequency of vibration is inverselyproportional to the length of air inside thebottle. Express your understanding of thisresonance phenomenon by filling in thefollowing table.




PhysicsTutorial


Lesson 1

Lesson 2

Lesson 3

Lesson 4

Lesson 5

Resonance

GuitarStrings

Open-EndAirColumns

Closed-End AirColumns


Resonance

The goal of Unit 11 of The PhysicsClassroom is to develop an understandingof the nature, properties, behavior, andmathematics of sound and to apply thisunderstanding to the analysis of music and

musical instruments. Thus far in this unit,applications of sound wave principles havebeen made towards a discussion of beats,musical intervals, concert hall acoustics,the distinctions between noise and music,and sound production by musicalinstruments. In Lesson 5, the focus will beupon the application of mathematicalrelationships and standing wave concepts tomusical instruments. Three general

categories of instruments will beinvestigated: string instruments (whichwould include guitar strings, violin strings,and piano strings), open-end air columninstruments (which would include the brassinstruments such as the flute and tromboneand woodwinds such as the saxophone andoboe),and closed-end air columninstruments (which would include theclarinet). A fourth category - vibrating

mechanical systems (which includes all thepercussion instruments) - will not bediscussed. These instrument categoriesmay be unusual to some; they are basedupon the commonalities among theirstanding wave patterns and themathematical relationships between thefrequencies which the instruments produce.

As was mentioned in Lesson 4, musical

instruments are set into vibrational motionat their natural frequency when a personhits, strikes, strums, plucks or somehowdisturbs the object. Each natural frequency




of the object is associated with one of themany standing wave patterns by which thatobject could vibrate. The naturalfrequencies of a musical instruments are

sometimes referred to as the harmonics of the instrument. An instrument can beforced into vibrating at one of its harmonics(with one of its standing wave patterns) if another interconnected object pushes itwith one of those frequencies. This isknown as resonance - when one objectvibrating at the same natural frequency of asecond object forces that second object intovibrational motion.

The word resonance comes from Latin andmeans to "resound" - to sound out togetherwith a loud sound. Resonance is a commoncause of sound production in musicalinstruments. In class, one of our models ofresonance in a musicalinstrument included theresonance tube (a hollowcylindrical tube) immersed in a

cylinder of water and forcedinto vibration by a tuning fork.The tuning fork was the objectwhich forced the air inside of the resonance tube intoresonance. As the tines of thetuning fork vibrated at theirown natural frequency, theycreated sound waves which impinged uponthe opening of the resonance tube. These

impinging sound waves produced by thetuning fork forced air inside of theresonance tube to vibrate at the samefrequency. Yet, in the absence of resonance, the sound of these vibrations isnot loud enough to discern. Resonance onlyoccurs when the first object is vibrating atthe natural frequency of the second object.So if the frequency at which the tuning forkvibrates is not identical to one of the

natural frequencies of the air column insidethe resonance tube, resonance will notoccur and the two objects will not soundout together with a loud sound. But the




resonance tube can be moved up and downwithin the water, thus decreasing orincreasing the length of the air column. Aswe have learned earlier, an increase in the

length of a vibrational system (here, the airin the tube) increases the wavelength anddecreases the natural frequency of thatsystem. Conversely, a decrease in thelength decreases the wavelength andincreases the natural frequency. So bymoving the resonance tube up and downwithin the water, the natural frequency of the air in the tube could be matched to thefrequency at which the tuning fork vibrates.

When the match is achieved, the tuningfork forces the air column inside of theresonance tube to vibrate at its own naturalfrequency and resonance is achieved. Andalways, the result of resonace is a bigvibration - that is, a loud sound.

Resonance was also modeled in class bythe demonstration with the famous "singingrod." A long hollow aluminum rod was held

by the teacher at its center. Being a trainedmusician, he/she reached in the rosin bagto prepare for the event. Then with greatenthusiasm, he/she slowly slid her handacross the length of the aluminum rod,causing it to sound out with a loud sound.This once more was an example of resonance. As the hand is slid across thesurface of the aluminum rod, slip-stickfriction between the hand and the rod

produces vibrations of the aluminum. Thevibrations of the aluminum forces the aircolumn inside of the rod to vibrate at itsnatural frequency. The match between thevibrations of the rod and one of the naturalfrequencies of the singing rod causesresonance. And always, the result of resonace is a big vibration - that is, a loudsound.







PhysicsTutorial


Lesson 1

Lesson 2

Lesson 3

Lesson 4

Lesson 5

Resonance

GuitarStrings

Open-EndAirColumns



Guitar Strings

A guitar string has a number of frequenciesat which it will naturally vibrate. Thesenatural frequencies are known as theharmonics of the guitar string. Asmentioned earlier, the natural frequency at

which an object vibrates at depends uponthe tension of the string, the linear densityof the string and the length of the string.Each of these natural frequencies orharmonics is associated with a standingwave pattern. The specifics of the patternsand their formation were discussed inLesson 4. For now, we will merelysummarize the results of that discussion.The graphic below depicts the standing

wave patterns for the lowest threeharmonics or frequencies of a guitar string.

The wavelength of the standing wave forany given harmonic is related to the lengthof the string (and vice versa). If the lengthof a guitar string is known, the wavelengthassociated with each of the harmonicfrequencies can be found. Thus, the length-wavelength relationships and the wave

equation (speed = frequency * wavelength)can be combined to make performcalculations predicting the length of stringrequired to produce a given natural




frequency. And conversely, calculations canbe performed to predict the naturalfrequencies produced by a known length of string. Each of these calculations requires a

knowledge of the speed of a wave in astring. The graphic below depicts therelationships between the key variables insuch calculations. These relationships willbe used to assist in the solution toproblems involving standing waves inmusical instruments.

To demonstrate the use of the aboveproblem-solving scheme, consider thefollowing problem and its detailed solution.

Practice ProblemThe speed of waves in a particular guitarstring is found to be 425 m/s. Determine

the fundamental frequency (1st harmonic)

of the string if its length is 76.5 cm.

The solution to the problem begins by firstidentifying known information, listing thedesired quantity, and constructing adiagram of the situation.

Given:

v = 425 m/s

L = 76.5 cm = 0.765 m

Find:

f 1 = ??

Diagram:

The problem statement asks us todetermine the frequency (f) value. From




the graphic above, the only means of finding the frequency is to use the waveequation (speed=frequency*wavelength)and knowledge of the speed and

wavelength. The speed is given, butwavelength is not known. If the wavelengthcould be found then the frequency could beeasily calculated. In this problem (and anyproblem), knowledge of the length and theharmonic number allows one to determinethe wavelength of the wave. For the firstharmonic, the wavelength is twice thelength. This relationship is derived from thediagram of the standing wave pattern (and

was explained in detail in Lesson 4). Therelationship, which works only for the firstharmonic of a guitar string, is used tocalculate the wavelength for this standingwave.

Wavelength = 2 * Length

Wavelength = 2 * 0.765 m

Wavelength = 1.53 m

Now that wavelength is known, it can becombined with the given value of the speedto calculate the frequency of the firstharmonic for this given string. Thiscalculation is shown below.



frequency = (425 m/s)/(1.53 m)

frequency = 278 Hz

Most problems can be solved in a similarmanner. It is always essential to take theextra time needed to set the problem up;take the time to write down the giveninformation and the requested information,and to draw a meaningful diagram.




Seldom in physics are two problemsidentical. The tendency to treat everyproblem the same way is perhaps one of

the quickest paths to failure. It is muchbetter to combine good problem-solvingskills (part of which involves the disciplineto set the problem up) with a solid grasp of the relationships among variables, than tomemorize approaches to different types of problems. To further your understanding of these relationships, examine the followingproblem and its solution.

To demonstrate the use of the aboveproblem-solving scheme, consider thefollowing problem and its detailed solution.

Practice Problem

Determine the length of guitar stringrequired to produce a fundamentalfrequency (1st harmonic) of 256 Hz. The

speed of waves in a particular guitarstring is known to be 405 m/s.

The solution to the problem begins by firstidentifying known information, listing thedesired quantity, and constructing adiagram of the situation.

Given:

v = 405 m/s

f 1 = 256 Hz

Find:

L = ??

Diagram:

The problem statement asks us todetermine the length of the guitar string.From the graphic above, the only means of

finding the length of the string is fromknowledge of the wavelength. But thewavelength is not known. However, thefrequency and speed are given, so one can




use the wave equation (speed =frequency*wavelength) and knowledge of the speed and frequency to determine thewavelength. This calculation is shown

below.


wavelength = speed/frequency

wavelength = (405 m/s)/(256 Hz)

wavelength = 1.58 m

Now that the wavelength is found, the

length of the guitar string can becalculated. For the first harmonic, thelength is one-half the wavelength . Thisrelationship is derived from the diagram of the standing wave pattern (and wasexplained in detail in Lesson 4); it may alsobe evident to you by looking at the standingwave diagram drawn above. Thisrelationship between wavelength andlength, which works only for the first

harmonic of a guitar string, is used tocalculate the wavelength for this standingwave.

Length = (1/2) * Wavelength

Length = (1/2) * Wavelength

Length = 0.791 m

If you have successfully managed theabove two problems, take a try at thefollowing practice problems. As youproceed, be sure to be mindful of thenumerical relationships involved in suchproblems. And if necessary, refer to thegraphic above.





PhysicsTutorial


Lesson 1

Lesson 2

Lesson 3

Lesson 4

Lesson 5

Resonance

GuitarStrings

Open-EndAirColumns



Open-End Air Columns

Many woodwind instruments consist of anair column enclosed inside of a hollow metaltube. Though the metal tube may be morethan a meter in length, it is often curvedupon itself one or more times in order to

conserve space. If the end of the tube isuncovered such that the air at the end of the tube can freely vibrate when the soundwave reaches it, then the end is referred toas an open end. If both ends of the tubeare uncovered or open, the musicalinstrument is said to contain an open-endair column. A variety of instrumentsoperate on the basis of open-end aircolumns; examples include the brass

instruments such as the flute and tromboneand woodwinds such as the saxophone andoboe. Even wind chimes and some organpipes serve as open-end air columns.

As has already been mentioned, a musicalinstrument has a set of natural frequenciesat which it vibrates at when a disturbance isintroduced into it. These natural frequenciesare known as the harmonics of the

instrument; each harmonic is associatedwith a standing wave pattern. In Lesson 4 of Unit 10, a standing wave pattern wasdefined as a vibrational pattern createdwithin a medium when the vibrationalfrequency of the source causes reflectedwaves from one end of the medium tointerfere with incident waves from thesource in such a manner that specific pointsalong the medium appear to be standing

still. If a sound wave is traveling through acylindrical tube, it will eventually come tothe end of the tube. The end of the tuberepresents a boundary between the




enclosed air in the tube and the expanse of air outside of the tube. Upon reaching theend of the tube, the sound wave willundergo partial reflection and partial

transmission. Inversion of the reflectedportion of the sound wave will occur only if the end of the tube is closed (and thus, actsas a fixed end). So for open-end aircolumns, the standing wave pattern whichresults is the outcome of the interference of an non-inverted reflected wave interferingwith an incident wave.

But how do the high-pressure

(compressions) and low-pressure(rarefactions) regions of a sound waveinterfere to produce a standing wavepattern? Suppose that a compression isintroduced into one end of the tube; thiscompression will reflect as a compression(i.e., no inversion). Now suppose that at theprecise moment that the reflection occurs, ararefaction is introduced into the originalend of the tube. Since a compression and a

rarefaction make up one-half of a wave, thisis the same as saying that there is one-half of a wave in the tube. If this is the case, thenewly introduced rarefaction willdestructively interfere with the reflectedcompression in the exact center of the tube.The destructive interference of a highpressure region (compression) and a lowpressure region (rarefaction) results innormal pressure region. So if one-half of a

wave is introduced into the open-end aircolumn, a standing wave consisting of apressure node in the center of the tube willbe established in the tube. Since there mustbe an alternating pattern of nodes and anti-nodes, then the open ends of the tube willalways have pressure antinodes. That is, thepressure at the open ends of an open-endair column are always oscillating between ahigh pressure and a low pressure. These

principles are depicted in the animationbelow.





The pressure plot in the animation abovedepicts the pressure on the ends of the aircolumn oscillating between a high pressureand a low pressure. The pressure in theexact center of the open-end air columnremains at normal pressure. For this reason,the standing wave pattern for thefundamental frequency (or first harmonic)for an open-end air column is shown in thediagram below.

One full pressure wave would start with a

compression region, lead into a normalpressure region, then lead into a rarefactionregion, back into a normal pressure region,and finish up when the next adjacentcompression region. From this description of a complete pressure wave and from acareful analysis of the two diagrams above,it is evident that there is one-half of apressure wave present in the length of theair column for the first harmonic.

The standing wave pattern for the secondharmonic of an open-end air column couldbe produced if another pressure node wasadded to the pattern. This would result in atotal of three pressure antinodes and twopressure nodes. This pattern is shown in thediagram below. Observe in the pattern thatthere is one full wave in the length of the aircolumn. That is twice the number of waves

in the first harmonic. For this reason, thefrequency of the second harmonic is twotimes the frequency of the first harmonic. 2001 The Physics Classroom and Mathsoft Education and




PhysicsTutorial


Lesson 1

Lesson 2

Lesson 3

Lesson 4

Lesson 5

Resonance

GuitarStrings

Open-EndAirColumns



Closed-End Air Columns

In the previous part of Lesson 5, theformation of a standing wave patterns in anopen-end instrument was discussed and themathematics of the harmonic frequenciesassociated with such standing wave patterns

was developed. This part of Lesson 5 willuse similar principles to develop thestanding wave patterns and associatedmathematics for closed-end instrument. Aclosed-end instrument is an instrument inwhich one of the ends of the metal tubecontaining the air column is covered. Anexample of an instrument which operates onthe basis of closed-end air columns is theclarinet. Some instruments which operate as

open-end air columns can be transformedinto closed-end air columns by covering theend opposite the mouthpiece with a mute.Even some organ pipes serve as closed-endair columns. As we will see the presence of the closed end on such an air column willeffect the actual frequencies which theinstrument can produce.

As has already been mentioned, a musical

instrument has a set of natural frequenciesat which it vibrates at when a disturbance isintroduced into it. These natural frequenciesare known as the harmonics of theinstrument; each harmonic is associatedwith a standing wave pattern. In Lesson 4 of Unit 10, a standing wave pattern wasdefined as a vibrational pattern createdwithin a medium when the vibrationalfrequency of the source causes reflected

waves from one end of the medium tointerfere with incident waves from thesource in such a manner that specific pointsalong the medium appear to be standing




still. If a sound wave is traveling through acylindrical tube, it will eventually come tothe end of the tube. The end of the tuberepresents a boundary between the

enclosed air in the tube and the expanse of air outside of the tube. Upon reaching theend of the tube, the sound wave willundergo partial reflection and partialtransmission. Inversion of the reflectedportion of the sound wave will occur only if the end of the tube is closed (and thus, actsas a fixed end). So for closed-end aircolumns, the standing wave pattern whichresults is the outcome of the interference of

an inverted reflected wave interfering withan incident wave.

But how do the high-pressure(compressions) and low-pressure(rarefactions) regions of a sound waveinterfere to produce a standing wavepattern? Suppose that a compression isintroduced into the open end of the tube;this compression will reflect as a rarefaction

(i.e., it will invert upon reflection off theclosed end). The rarefaction will thansubsequently return towards the open endof the tube to interfere with other parts of the wave which are heading the oppositedirection. Now suppose that at the precisemoment that the reflected rarefactionreaches the open end of the tube, acompression is introduced into the openend. The original disturbance has now

traveled twice the length of the tube in thetime that it has taken to introduce one-half of a wave cycle. For this reason, there isone-fourth of a wave in any given length of the tube. If this is the case (length = one-fourth wavelength), the newly introducedrarefaction will constructively interfere withthe reflected rarefaction at the open end of the tube. The constructive interference of alow pressure region (rarefaction) and a low

pressure region (rarefaction) results in verylow pressure region. The same reasoningcan be used to explain how the introductionof a rarefaction into the open end will result




in constructive interference with the nextcompression at that same open end.Constructive interference will always occurat the open end of any air column. So if

one-fourth of a wave is introduced into aclosed end air column, a standing wavepattern consisting of a pressure anti-node atthe open end of the air column will beestablished in the tube. Since there must bean alternating pattern of nodes and anti-nodes, then the closed end of the tube willalways have pressure nodes. That is, thepressure at the open end of a closed end aircolumn are always oscillating between a

high pressure and a low pressure while theclosed end is always maintained at a normalpressure level. These principles are depictedin the animation below.

The pressure plot in the animation abovedepicts the pressure on the open end of theair column oscillating between a highpressure and a low pressure. The pressureon the closed end of the closed-end aircolumn remains at normal pressure. Thiswill always be the case for any givenharmonic. Since the closed end acts as afixed end which prevents the oscillation of air, the closed end remains at normalpressure; the close ends are alwayspressure nodes. On the other hand, theopen ends are always pressure anti-nodes;since air is free to oscillate into and out of the tube at the open ends, the pressure atthese ends are oscillating back-and-forthfrom high pressure to low pressure. For thisreason, the standing wave pattern for thefundamental frequency (or first harmonic)




for a closed-end air column is shown in thediagram below.

One full pressure wave would start with acompression region, lead into a normalpressure region, then lead into a rarefactionregion, back into a normal pressure region,and finish up when the next adjacentcompression region. From this description of a complete pressure wave and from acareful analysis of the two diagrams above,

it is evident that there is one-fourth of apressure wave present in the length of theair column for the first harmonic.

The fundamental frequency is the lowestpossible frequency which any instrumentcan play; it is sometimes referred to as thefirst harmonic of the instrument. The second

harmonic of any instrument always has afrequency which is twice the frequency of the first harmonic. The fourth harmonic of any instrument always has a frequencywhich is four times the frequency of the firstharmonic. As we will see, strange patternresults for a closed-end air column. Just asfor all the instruments, the next harmonicfor a closed end air column is the harmonicwhich has one more node. And just as for all

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