Chapter21 Sound Waves

7/23/2019 Chapter21 Sound Waves

http://slidepdf.com/reader/full/chapter21-sound-waves 1/8

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

2

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

3

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)

4

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.

7

How they work

A sound wave ismeasured in hertz (Hz) =>

vibration/second

These are High andLow frequency waves,they show thedifference betweenthe two.

8



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.

9

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

10

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.)

11

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.

12

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.

13


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

14


3. The general equation for the pressurefluctuations:

P(x, t) = P o sin(kx - t) orP = P o sin ( t kx + /2)

15

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.

16



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 .

17

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.

18

21.2 Sources of Sound: Vibrating AirColumns

Wind instrumentscreate sound

through standing waves in a columnof air.

19

21.2 Sources of Sound: Vibrating Stringsand Air Columns

20

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

21

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:

22

Vibrating Membrane


23


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:

24

2 2

medium wave m

Power PI = = =

Area A




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:

25

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)

26

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

27 28

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

29

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

30

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.



33


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

34

beat 1 2 f f f

CP 535


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

35

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)

36

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)

37

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)

38

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)

39

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.

40

o

o ss

v v

f f v v

Source (s) Observer (o)



41

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.

42

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?

43

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

44

45

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

46

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

/

47

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?

48

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



49


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'

50


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'

51

(5)


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'

52

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


The frequency of sound wave heard by O1

increases as given by,

53

(6)

T v

vv f

s''

f v f v

v

s/

svv

v f f '


With respect to observer2, the wavelength ofsound increases, where

vsT.The frequency of sound wave heard by O2 decreases as given by,

54

svv

v

f f '


Combining Equations (6) and (7), we have

55

(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,

56

(Moving source and observer)

(9)

svvvv f f 0'




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.

57

(9)

svv

vv f f 0'


Approaching observer,receding source

If v o> v s , observedfrequency increasesIf v o< v s , observedfrequency decreases

58

svv

vv f f 0'

svv

vv f f 0'

Receding observer,receding source

Decrease inobserved frequency


Approaching observer,approaching source

Observed frequencyincreases

59

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'

60

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?

61

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)]

62

Chapter21 Sound Waves

Documents

Transcript of Chapter21 Sound Waves