Physics CPhysics CPhysics CPhysics C

Chapter Chapter Chapter Chapter 17 17 17 17

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M.Sc.( master degree) in Theoretical Physics,M.Sc.( master degree) in Theoretical Physics,M.Sc.( master degree) in Theoretical Physics,M.Sc.( master degree) in Theoretical Physics,

Electromagnetic Waves (Optical Science) ,Electromagnetic Waves (Optical Science) ,Electromagnetic Waves (Optical Science) ,Electromagnetic Waves (Optical Science) ,

Islamic University of Gaza (Gaza, Palestine).Islamic University of Gaza (Gaza, Palestine).Islamic University of Gaza (Gaza, Palestine).Islamic University of Gaza (Gaza, Palestine).

Chapter TwoSound waves

17.1 speed of sound waves17.1 speed of sound waves

17.2 periodic sound waves

17.3 Intensity of Periodic Sound Waves

17.4 The Doppler Effect

* Sound waves* Sound waves* Sound waves* Sound waves are the most common example of longitudinal waves.

introductionintroductionintroductionintroduction

* * * * The speed of soundThe speed of soundThe speed of soundThe speed of sound waves depends on the propertiespropertiespropertiesproperties of the medium.

****Sound waves are divided into three types that cover Sound waves are divided into three types that cover Sound waves are divided into three types that cover Sound waves are divided into three types that cover

different frequency ranges:different frequency ranges:different frequency ranges:different frequency ranges:

(1) Audible wavesAudible wavesAudible wavesAudible waves :::: lie within the range of sensitivity of the human ear.(1) Audible wavesAudible wavesAudible wavesAudible waves :::: lie within the range of sensitivity of the human ear.

((((2222)))) Infrasonic wavesInfrasonic wavesInfrasonic wavesInfrasonic waves : : : : have frequencies below the audible range, have frequencies below the audible range, have frequencies below the audible range, have frequencies below the audible range,

Elephants can use infrasonic waves to communicate with each Elephants can use infrasonic waves to communicate with each Elephants can use infrasonic waves to communicate with each Elephants can use infrasonic waves to communicate with each

other, even when separated by many kilometers.other, even when separated by many kilometers.other, even when separated by many kilometers.other, even when separated by many kilometers.

((((3333)))) Ultrasonic waves :Ultrasonic waves :Ultrasonic waves :Ultrasonic waves :

((((3333) ) ) ) Ultrasonic wavesUltrasonic wavesUltrasonic wavesUltrasonic waves : : : : have frequencies above the audible range.have frequencies above the audible range.have frequencies above the audible range.have frequencies above the audible range.

17171717....1 1 1 1 Speed of Sound Waves:Speed of Sound Waves:Speed of Sound Waves:Speed of Sound Waves:

** Before the piston is moved, the gas is

undisturbed and of uniform density, as

represented by the uniformly shaded

region in Figure (a). region in Figure (a).

**When the piston is suddenly pushed to

the right Figure (b). the gas just in front

of it is compressed (shaded regionshaded regionshaded regionshaded region).

**the pressure and density in this region

are now higher than others.

** When the piston comes to

rest Figure c, the compressed

region of the gas continues to

move to the right,

corresponding to a longitudinal

pulse traveling through the tube

with speed vvvv....with speed vvvv....

*** Now the speed of sound waves in a medium depends on the

compressibilitycompressibilitycompressibilitycompressibility and densitydensitydensitydensity of the medium.

**If the medium is a liquid or a gas and has a bulk modulus B and density then the speed of sound waves in medium is ρ

ρB

v =

µT

v =

Remark: Remark: Remark: Remark: this speed as the speed of transverse waves on

a string,

In fact, the speed of all mechanical wavesspeed of all mechanical wavesspeed of all mechanical wavesspeed of all mechanical waves follows an expression In fact, the speed of all mechanical wavesspeed of all mechanical wavesspeed of all mechanical wavesspeed of all mechanical waves follows an expression

of the general form

propertyinertial

propertyelasticv =

The speed of sound also depends on the temperature of the medium.

For example: the speed of sound in air given byFor example: the speed of sound in air given byFor example: the speed of sound in air given byFor example: the speed of sound in air given by

C

Tsmv

oc

2731/331 +=

where 331m/s is the speed of sound in air at 0°C,

Tc is the air temperature in degrees Celsius.

Using this equation the speed of sound at is v = m/sUsing this equation the speed of sound at Tc =20 °C is v = 343 m/sRemark : Remark : Remark : Remark : the way to estimate the distance (Km) to a thunderstorm is

number of seconds between seeing the flash of lightning and hearing

the thunder divided by 3 . (explain ??)(explain ??)(explain ??)(explain ??)

17171717....2 2 2 2 Periodic Sound Waves: Periodic Sound Waves: Periodic Sound Waves: Periodic Sound Waves:

** the gas is compressed and thus the

density and pressure are above their

equilibrium values.

** This compressed region, called a

compressioncompressioncompressioncompression

** The areas of low pressure regions,

called rarefactionsrarefactionsrarefactionsrarefactions

** As the piston oscillates

sinusoidally, regions of compression

and rarefaction are continuously set

up.

the position of a small element relative to its equilibrium position is

given by )cos(),( max tkxstxs ω−=

Where SSSSmaxmaxmaxmax is the maximum position of the element relative to equilibrium

(displacement amplitude).(displacement amplitude).(displacement amplitude).(displacement amplitude).

The type of wave in the figure is longitudinal wavelongitudinal wavelongitudinal wavelongitudinal wave

(1)

P∆

The type of wave in the figure is longitudinal wavelongitudinal wavelongitudinal wavelongitudinal wave

The variation in the gas pressure measured from the equilibrium

value is also periodic. And given by,

)sin(max tkxPP ω−∆=∆ (2)

maxP∆where is the pressure amplitude

maxmax svP ωρ=∆

which is the maximum change in pressure from the equilibrium value

Equations (1) and (2) shows that the pressure wave is 90°out of

phase with the displacement wave.

(3)

RemarksRemarksRemarksRemarks::::

1) the pressure variation is a

maximum when the displacement

from equilibrium is zero.

2) the displacement from equilibrium

is a maximum when the pressureis a maximum when the pressure

variation is zero.

17171717....3 3 3 3 Intensity of Periodic Sound Waves :Intensity of Periodic Sound Waves :Intensity of Periodic Sound Waves :Intensity of Periodic Sound Waves :

To evaluate the rate of energy transfer for the sound wave, we

shall evaluate the kinetic energy of this element of air, which is

In the preceding chapter, we showed that a wave traveling on a taut

string transports energy. The same concept applies to sound waves.

Consider an element of air of mass m and width x in front of a

piston oscillating with a frequency , as shown in Figure

∆

ω∆

shall evaluate the kinetic energy of this element of air, which is

undergoing simple harmonic motion.

Firstly the kinetic element of one element is given by 2

2

1vdmdk =

[ ] )sin()cos(),(),( maxmax tkxStkxSx

txsx

txv ωωω −−=−∂∂=

∂∂=

Now, we must find the speed of the element dm

(1)

(2)

Also the element is given by dmAlso the element is given by dm

dxAdm ρ= (3)

Substituting from Eqn (2) and Eqn (3) into Eqn (1) to get

[ ]

0)(sin2

1

)(sin2

1)sin(

2

1

222max

222max

2max

==

−=−−=

tatdxkxSAdk

tkxdxSAtkxSdxAdk

ωρ

ωωρωωρ

(4)

22

1

4

)2sin(

22

1

)2cos(2

1

2

1

2

1)(sin

2

1

22max

0

22max

0

22max

0

222max

λωρωρ

ωρωρ

λ

λ

λλ

λ

SAk

kxxSAk

dxkxSAdxkxSAk

=

−=

−== ∫∫

Integrating the both sides of Eqn (4) to get

That is, the kinetic energy in one wave length is given by

λωρλ22

max4

1SAk =

and the potential energy in one wave length is given by

λωρλ22

max4

1SAU =

and the total energy in one wave length is given by

λωρλ22

max2

1SAE =

The rate of energy transfer (power)(power)(power)(power) is given by

vSAT

SAT

EP 22

max22

max 2

1

2

1 ωρλωρλ ===

vSAP 22max2

1 ωρ= (5)

We define the intensity of a wave, or the power per unit area, to beWe define the intensity I of a wave, or the power per unit area, to be

the rate at which the energy being transported by the wave transfers

through a unit area A perpendicular to the direction of travel of the

wave:

A

PI = (6)

A

vSA

A

PI

22max2

1 ωρ==

From Eqn (5) into Eqn (6) to get

vSI 221 ωρ=

Finally, the intensity can be written as

(7)vSI 22max2

ωρ= (7)

RemarkRemarkRemarkRemark:::: we see that the intensity of a periodic sound wave is

proportional to the squaresquaresquaresquare ofofofof thethethethe displacementdisplacementdisplacementdisplacement amplitudeamplitudeamplitudeamplitude and to

the squaresquaresquaresquare ofofofof thethethethe angularangularangularangular frequencyfrequencyfrequencyfrequency (as in the case of a periodic

string wave).

The intensity can also be written in terms of the pressure amplitudepressure amplitudepressure amplitudepressure amplitude

v

PI

ρ2

2max∆

=

That is Eqn (7) becomes

(8)

Now consider a point source emitting sound waves equally in all directions.Now consider a point source emitting sound waves equally in all directions.

We identify an imaginary sphere of radius r centered on the source.

The average power emitted by the source must be distributed

uniformly over this spherical surface of area .

avP24 rπ

24 r

PI av

π=

Hence, the wave intensity at a distance r from the source is

(9)

Sound Level in DecibelsSound Level in DecibelsSound Level in DecibelsSound Level in Decibelsthe sound level (Greek beta) is defined by the equationβ

=

oI

Ilog10β

The constant Io is the reference intensity, taken to be at the threshold of

hearing212 /101 mwI o

−×=hearing

β is is is is measured in decibelsdecibelsdecibelsdecibels (dB).

Remark: Remark: Remark: Remark:

1)1)1)1) The sound level of the threshold of pain is The sound level of the threshold of pain is The sound level of the threshold of pain is The sound level of the threshold of pain is

2222) ) ) ) The sound level of the threshold of hearingThe sound level of the threshold of hearingThe sound level of the threshold of hearingThe sound level of the threshold of hearing

is is is is

2/1 mwI = dB120=β212 /101 mwI o

−×=

dB0=β