chapter21 sound waves
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7/23/2019 Chapter21 Sound Waves
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1
Chapter 21. Sound waves
Content
21.1 Propagation of sound waves
21.2 Sources of sound
21.3 Intensity of sound
21.4 Beat
21.5 Doppler effect
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objectives
a) explain the propagation of sound waves in airin terms of pressure variation and
displacementb) interpret the equations for displacement, y = yo sin ( t kx), and pressure, p = po sin ( t kx + /2)
c) use the standing wave equation to determinethe positions of nodes and antinodes of astanding wave along a stretched string
d) use the formula v = (T/ )1/2 to determine thefrequencies of the sound produced bydifferent modes of vibration of the standing waves along a stretched string
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objectives
describe, with appropriate diagrams, thedifferent modes of vibration of standing waves
in air columns, and calculate the frequenciesof sound produced, including thedetermination of end correctiondefine and calculate the intensity level ofsounduse the principle of superposition to explainthe formation of beatsuse the formula for beat frequency, f = f 1 f 2describe the Doppler effect for sound, and usethe derived formulae (for source and/orobserver moving along the same line)
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What are sound waves?
A mechanical wave that vibrates amedium (like air or water) with differentfrequencies.These frequencies are then picked up byour ears.They are created through a variety ofinteractions, but all are mechanical
(Physical).5
When we use Sound WavesMusic ties into Sound waves andfrequencies.Each note has a different frequency.
We talk through sound waves, and applymeaning to certain sounds.Dolphins and bats use sound wave (sonar ). Dolphins use it to communicate, like a
language, and bats use them to fly due topoor eye sight.6
How they work
Sound waves travel in alongitudinal way (verticalfashion), as shown by thetuning fork in the picture.
The sound vibrates themedium between the
whatever is straight infront of it.
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How they work
A sound wave ismeasured in hertz (Hz) =>
vibration/second
These are High andLow frequency waves,they show thedifference betweenthe two.
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How they work
The periods, T betweenthe waves categorizetheir frequencies, f aslow or high. f 1/TThe higher frequency has
a smaller amount of time between waves, while thelower frequency has alonger amount of time.
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Frequencies
Interval Frequency Ratio Examples
Octave 2:1 512 Hz and 256 Hz
Third 5:4 320 Hz and 256 Hz
Fourth 4:3 342 Hz and 256 Hz
Fifth 3:2 384 Hz and 256 Hz
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This chart explains sound waves pertain to
musicmake music. For instance, raising a note anoctave would require multiplying the base noteby 2 (take a low c, with frequency of 261.5, toraise it an octave: has frequency 523.)
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Frequency
This table shows the value in hertz ofcertain notes (rounding applies).
Note C C# D D# E F F# G G# A A# B C C# D
Octave 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2
Hz 262 278 294 311 330 349 370 392 415 440 466 494 523 554 587
Sound
A longitudinal traveling waveProduced by vibrations in a medium
The disturbance is the local change in pressure generated by the vibrating objectIt travels because of the molecularinteractions.
The region of increased pressure (compared tothe normal pressure) is called condensationThe region of lower pressure is calledrarefaction.
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Sound
The maximum increase in pressure (DPm) isthe amplitude of the pressure wave.(measurable)
frequency : 20Hz to 20kHz.Pressure waves below 20 Hz are calledinfrasonic wavesPressure waves over 20kHz are calledultrasonic waves.
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21.1 Propagation of sound waves
1. The propagation of sound waves occurs dueto the oscillations of individual particles withthe medium producing traveling waves ofpressure fluctuations
2. The general form of particle oscillation y (x, t) = y o cos(kx - t) or y = yo sin ( t kx)where yo is the magnitude of the particle
displacement
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21.1 Propagation of sound waves
3. The general equation for the pressurefluctuations:
P(x, t) = P o sin(kx - t) orP = P o sin ( t kx + /2)
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21.2 Sources of Sound
Musical instruments produce sounds in various ways vibrating strings, vibrating membranes, vibrating metal or wood shapes, vibrating air columns.
The vibration may be started by plucking,striking, bowing, or blowing.The vibrations are transmitted to the air and then to our ears.
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21.2 Sources of Sound: Vibrating Strings
The strings on a guitarcan be effectivelyshortened by fingering,raising thefundamental pitch.The pitch of a string of
a given length can alsobe altered by using astring of differentdensity .
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21.2 Sources of Sound: Vibrating Strings
A piano uses bothmethods to cover itsmore than seven-octave range: thelower strings (at
bottom) are bothmuch longer andmuch thicker thanthe higher ones.
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21.2 Sources of Sound: Vibrating AirColumns
Wind instrumentscreate sound
through standing waves in a columnof air.
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21.2 Sources of Sound: Vibrating Stringsand Air Columns
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A tube open at both ends (most windinstruments) has pressure nodes, and therefore
displacement antinodes, at the ends.
21.2 Sources of Sound: Vibrating Stringsand Air Columns
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A tube closed at one end (some organ pipes) hasa displacement node (and pressure antinode) at
the closed end.
21.2 Sources of Sound: VibratingMembrane
A piece of elastic membrane can vibrate in themodes as shown in the figure below:
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Vibrating Membrane
21.3 Intensity of sound
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21.3 Intensity of sound
Waves transport energy withouttransporting mass. The amount of energytransported per second is the power (P)of the wave (in W)
Intensity is a measure of powertransmitted by a wave per unit area:
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2 2
medium wave m
Power PI = = =
Area A
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21.3 Intensity of sound
The energy transmission (power) isdetermined by the source.The power is distributed (spreads) in alldirections. Far away from the source, thepower is spread over a greater area.For a point source, intensity decreasesinversely with the square of the distancefrom the source:
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2
P PI(r) = =
A 4 r
Loudness & Decibels
1. The human does not perceive sound intensitylinearly but rather logarithmically
Perceived Loudness, I perceived log (I actual)2. The average minimum perceivable sound
intensity:I o
-12 W/m2
3. The decibel scale was been developed to
ear perception (intensity level, ):= (10 dB). log(I /I 0) = (10 dB). log(I + 12)
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21.4 Beats
1. When 2 sound waves the resultant wavepattern exhibits both constructive and
destructive interference.2. When the amplitudes of the 2 waves are
similar but the frequencies are slightlydifferent then:a. The frequency of the resultant wave is
roughly the average frequency of the 2 waves
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21.4 Beats
2. When the amplitudes of the 2 waves are similar butthe frequencies are slightly different then:
a. The combined effect of interferenceproduces periodic rises and drops inloudness called beats
b.The frequency of the beats (f beat) is equal tothe difference between the 2 soundfrequencies: f beat = f 1 - f 2
3. Musicians often tune their musicalinstruments by listening to beat frequency
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21.4 BeatThe superposition of 2 sound waves:
f wave1=159.2 Hz f wave1=148.0 Hz
The resulting beat frequency:
f beat= f wave1 - f wave2
= 159.2 Hz - 148.0 Hz = 21.2 Hz
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21.4 Beat When two sound wavesof different but nearlyequal frequency ( f 1 and
f 2) superimpose, we anintensity variation at thedifference frequencyThe intensity variation iscalled beatsThe beat frequency isequal to the differencefrequency | f 1 - f 2|
1 beat
Used to tune musical
instruments to same pitch
31CP 535
21.4 Beat
Superimpose oscillations of equal amplitude,but different frequencies
Modulation of amplitudefrequency of pulses is | f1- f2 |
Oscillation at theaverage
frequency
Notexaminable
1 2
1 2 1 2
1 2 1 2
sin(2 ) sin(2 )
( ) ( )2 sin(2 )cos(2 )
2 2
( ) ( )2 cos(2 ) sin(2 )
2 2
A f t A f t
f f f f A t t
f f f f A t t
32CP 535
21.4 Beat interference in time
Consider two sound sources producing audiblesinusoidal waves at slightly different frequencies f 1and f 2. What will a person hear ? How can a pianotuner use beats in tuning a piano? If the two waves atfirst are in phase they will interfere constructivelyand a large amplitude resultant wave occurs which
will give a loud sound. As time passes, the two wavesbecome progressively out of phase until they interferedestructively and it will be very quite. The waves thengradually become in phase again and the patternrepeats itself. The resultant waveform shows rapidfluctuations but with an envelope that various slowly.
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21.4 Beat interference in time
The frequency of the rapid fluctuations is theaverage frequencies =
The frequency of the slowly varying envelope =
1 2
2
f f
1 2
2
f f
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beat 1 2 f f f
CP 535
21.4 Beat interference in time
Since the envelope has two extreme values ina cycle, we hear a loud sound twice in onecycle since the ear is sensitive to the square ofthe wave amplitude.The beat frequency is
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0
10
20
30
40
50
60
0 0.05 0.1 0.15 0.2 0.25
time
CP 535
f 1 = 100 Hz f 2 = 110 Hz f rapid = 105 Hz T rapid = 9.5 ms f beat = 10 Hz T beat = 0.1 s (loud pulsation every 0.1 s)
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0
10
20
30
40
50
60
0 0.05 0.1 0.15 0.2 0.25
time
f =100
f = 120
beats
CP 535
f 1 = 100 Hz f 2 = 120 Hz f rapid = 110 Hz T rapid = 9.1 ms f beat = 20 Hz T beat = 0.05 s (loud pulsation every 0.05 s)
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0
10
20
30
40
50
60
0 0.05 0 .1 0.15 0.2 0 .25
time
f =100
f = 104
beats
CP 535
f 1 = 100 Hz f 2 = 104 Hz f rapid = 102 Hz T rapid = 9.8 ms
f beat = 4 Hz T beat = 0.25 s (loud pulsation every 0.25 s)
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One might wonder why the siren on a movingambulance seems to produce sound with ahigher pitch when it passes an observer anddecreases when it recede the observer.Is this simply because of the relative distancebetween the observer and the ambulance(sound)?Or is it because of the loudness of the soundproduced by the siren?
21.5 Doppler effect
21.5 Doppler effect
Christian Johann Doppler(1803-1853)
Studied motion relatedfrequency changes (1842)
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o
o ss
v v
f f v v
Source (s) Observer (o)
21.5 Doppler effect
Doppler effect is the change in frequency ofa wave (or other periodic event) foran observer moving relative to its source.
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o
o ss
v v
f f v v
Source (s) Observer (o)
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21.5 Doppler effect
of waves and the observer are approachingeach other, the sound heard by the observerbecomes higher in pitch, whereas if the sourceand observer are moving apart the pitchbecomes lower.
For the sound waves to propagate it requires amedium such as air, where it serves as a frameof reference with respect to which motion ofsource and observer are measured.
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21.5 Doppler effect
Applications: police microwave speed unitsspeed of a tennis ballspeed of blood flowing through an arteryheart beat of a developing fetousburglar alarms
sonar ships & submarines to detectsubmerged objectsdetecting distance planetsobserving the motion of oscillating stars.
21.5 Doppler effect
Consider source ofsound at frequency f s,
moving speed vs,observer at rest (vo = 0)Speed of sound v What is frequency f o heard by observer?
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21.5 Doppler effect
On right - sourceapproaching source catching up on waves
wavelength reducedfrequency increasedOn left - source receding source moving away from
waves wavelength increasedfrequency reduced
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SITUATION 1 Stationary Source andObservers (NO DOPPLER EFFECT)
A stationary sound source Semits a spherical wavefronts of
v relative to the medium air.
In time t, the wavefronts move adistance vt toward theobservers, O1 & O2.
The number of wavelengthsdetected by the observer infront
and behind the source are thesame and equal to vt
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SITUATION 1 Stationary Source andObservers (NO DOPPLER EFFECT)
Thus, the frequency f heard by both stationaryobservers is given by,
f - frequency of sound source v - speed of sound wavest - time
- wavelength
v
t
vt f
/
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21.5 Doppler effect
What if both of the observers in figure 1 aremoving, is there any change in the frequencyand wavelength of the source?
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SITUATION 2 Stationary Source;Moving Observers
Observer 1 moves a distance vot toward the source at speed vo
We had known earlier that wavefronts also move at speed v towards O1 in time t at distance vt.
The distance traveled by the wavefronts with respect to O1 becomes vt + vOt.
The number of wavelengthsintercepted by O1 at this distanceis (vt + v0t
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SITUATION 2 Stationary Source;Moving Observers
This shows that there is an increase in thefrequency heard by O1 as it goes nearer to thesound source as given by,
(2)
Since = v/ f , then(3)
00 /)('
vv
t
t vvt f
v
vv f f 0'
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SITUATION 2 Stationary Source;Moving Observers
If observer 2 moves awayfrom the sound source,the distance traveled bythe wavefronts withrespect to O2 in time t, is
vt vot.Consequently, there
would be a decrease in thefrequency heard by O2 asgiven by,
(4)
v
vv f f 0'
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(5)
SITUATION 2 Stationary Source;Moving Observers
In these situations only the frequency heardby the observers changes due to there motionrelative to the source.However the wavelength of sound remainsconstant.
v
vv f f 0'
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SITUATION 3 Moving Source;Stationary Observers
As the source moves adistance vS T (T=1/f
period of wave) towardO1 there is a decrease inthe wavelength of soundby a quantity of vsT.The shortened
vsT
SITUATION 3 Moving Source;Stationary Observers
The frequency of sound wave heard by O1
increases as given by,
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(6)
T v
vv f
s''
f v f v
v
s/
svv
v f f '
SITUATION 3 Moving Source;Stationary Observers
With respect to observer2, the wavelength ofsound increases, where
vsT.The frequency of sound wave heard by O2 decreases as given by,
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svv
v
f f '
SITUATION 3 Moving Source;Stationary Observers
Combining Equations (6) and (7), we have
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(8)
(7)
(6)
svv
v
f f '
svv
v f f '
svv
v f f '
SITUATION 4 Moving Source andObserver
From the equations (5) and (8), we can nowderive the equation of general Doppler Effectby replacing f in equation (5) with ofequation (8). This result to,
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(Moving source and observer)
(9)
svvvv f f 0'
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SITUATION 4 Moving Source andObserver
The ± signs correspond to the direction of thesource or observer when they are moving
relative to the other. These would determine whether there is an increase or decrease on thefrequency heard by the observer during themotion.
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(9)
svv
vv f f 0'
SITUATION 4 Moving Source andObserver
Approaching observer,receding source
If v o> v s , observedfrequency increasesIf v o< v s , observedfrequency decreases
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svv
vv f f 0'
svv
vv f f 0'
Receding observer,receding source
Decrease inobserved frequency
SITUATION 4 Moving Source andObserver
Approaching observer,approaching source
Observed frequencyincreases
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Receding observer,approaching sourceIf v o> v s , observedfrequency decreasesIf v o< v s , observedfrequency increases
svv
vv f f 0'
svv
vv f f 0'
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Problem
A train has a whistle, which emits a 400 Hzsound. You are stationary and you hear the
whistle, but the pitch is 440 Hz. How fast istrain moving towards or away from you?
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Solution:The pitch is higher, so the train is movingtowards you.Its speed relative to you is found fromf = f 0 v/(v-v s). We have(v-v s) = f 0 v/f = (400/s)(330 m/s)/(440/s)= 300m/s.Therefore v s = 330m/s 300m/s
= 30m/s
Summary: Sound waves
Propagation
Sources of sound
Intensity level
Beat
Doppler effect
y = yo sin ( t kx)
p = po sin ( t kx + /2)
Open ends tube: Ln= n /2, f n = nv/2L
One Closed End tube: Ln = n /4, f n = nv/4L
I = P/4 r2
= (10 dB). log(I /I 0)
f beat =| f 1 - f 2 |
f f [(v v0)/(v vs)]
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