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```					AL Physics/Wave Motion                                                        K.W. LAM

ACOUSTICS
1. Sound Waves

   longitudinal
   mechanical
   requires a material medium for propagation
   produced by a vibrating object which superimposes on the movement of the particles of
the medium (e.g. air)

Sound audible to human, Frequency: 20 Hz to 20 kHz (most sensitive 1 to 3 kHz)
Speed:      about 330 m s-1 in air
Wavelength: 17 m to 17 mm

Sound waves of frequency lower than 20 Hz are called subsonic waves
while those greater than 20 kHz are called ultrasonic waves.

Consider a sound is emitted by a vibrating string. Sound waves push the neighbouring air
molecules so that compression and rarefaction are created.

The lines in the figure below represent the positions of the air molecules.

Acoustics                                    Page 1                               Sep 2007
AL Physics/Wave Motion                                                                 K.W. LAM

1.1 Pressure and Displacement in Sound Waves

As the air molecules oscillate to and fro about their undisturbed positions, compressions
(C) and rarefactions (R) travel to the right.

(A) Displacement

 Air at the centres of compressions and rarefactions has zero displacement.
 While air at the mid-point between consecutive compression and rarefaction has
maximum displacement

Note:
We usually take the right direction to be positive and left direction to be negative.

Acoustics                                       Page 2                                     Sep 2007
AL Physics/Wave Motion                                                              K.W. LAM

(B) Velocity

 Velocity of air at the centres of compressions and rarefactions are maximum.
(Since a particle undergoing SHM at equilibrium position has maximum
velocity)
 Velocity of air at the mid-point between consecutive compression and
rarefaction is zero. (Since a particle undergoing SHM at maximum displacement
has zero velocity)
 At compression  +ve velocity (to the right)
At rarefaction  -ve velocity (to the left)

(C) Pressure

 At the centre of compression
 pressure is maximum (higher than normal pressure)
 At the centre of rarefaction
 pressure is minimum (lower than normal pressure)

Q: What is the pressure at the mid-point between consecutive compressive and
rarefaction?

Note:
1. The variation of air density is similar to that of pressure.
2. The variation of pressure is found to have a phase difference of 90o with the
variation of displacement. (i.e. displacement curve _________ pressure curve
by 90o)

Related AL Question: [96/IIB/2(a)]

1.2 The Decibel

1.21 Relationship between Intensity and Loudness

(A) Intensity

Intensity is defined as the rate of flow of energy through a unit area perpendicular
to the direction of travel of the sound at the place in question.

energy
Intensity =
area  time
power
=
area
p
or     I =                           (Unit: Wm-2)
A
The intensity I at a distance r from a point source is

Acoustics                                          Page 3                              Sep 2007
AL Physics/Wave Motion                                                                 K.W. LAM

p
I =
4 r 2

1
So         I                 (sound intensity obeys inverse square law)
r2

Intensity of a sound wave is also directly proportional to the square of the
amplitude of the vibration of particles.
i.e.    I  A2

Note:
1. The absorption of sound in the surrounding medium is neglected. Actually the
wave energy is gradually absorbed by the medium’s molecules due to
attenuation.
2. Intensity is an absolute physical quantity. It ONLY depends on :
 the distance r between the listenser and the sound source;
 the amplutde A of the sound wave.

(B) Loudness

Loudness is a subjective sensation.
It depends on
1. the intensity of the sound
2. the sensitivity of the listener’s ears

Note:
Response of human ear varies with frequency of sound. The human ear is most
sensitive around 1 to 3 kHz.

1.22 Intensity Level

The ear responds to the ratio of the power and not to their difference.

e.g. change in loudness when the power of a sound increases from 0.1W to 1W =
increase from 1W to 10W.

So it is more convenient to compare the logarithm.

1           10
Now          log10        = log10    =1
0 .1          1

Acoustics                                        Page 4                                  Sep 2007
AL Physics/Wave Motion                                                           K.W. LAM

If a power change from P1 to P2 ,

P2
change in Intensity level h = 10 log10
P1

(Unit: Decibel, dB)

Example

(a)      Calculate the change in intensity level when the power of a sound increases from
100mW to 200mW.
(b)      Calculate the change in intensity level when the power of a sound increases to a
value
(i) double
(ii) 10 times
(iii) 100 times
(iv) 1000 times
of its original value.

Solution:

(a)

(b)

We can also use intensity of a sound to define the intensity level h.

Intensity level of a sound source is its intensity relative to some agreed ‘zero’ intensity
level.

I
Intensity level h = 10 log10
I0

where I0  10-12 Wm-2 is the minimum intensity detectable by a normal human ear.
I
So when intensity = I0, intensity level h = 10 log10 0 = 10 log10 1 = 0 dB. And 0 dB is
I0
taken as the threshold of hearing (or threshold of audibility). i.e., the sound the normal
human ear can just detect.

Acoustics                                       Page 5                               Sep 2007
AL Physics/Wave Motion                                                             K.W. LAM

Sound of intensity level 120 dB will produce a painful sensation to a normal human ear
and this value is known as threshold of pain (or threshold of feeling).

The difference in intensity level h of two sound waves of intensities I1 and I2,

h = h2 – h1
I2           I
= 10 log10      - 10 log10 1
I0           I0
I2 I1
= 10 log10 (     / )
I0 I0
I2
h = 10 log10
I1

Related AL Question: [92/II/2(b)(ii)]

Each of the curve below shows the loudness of a sound depends on its frequency.

Each curve corresponds to the same loudness to a normal human ear.

It is found that the human ear is quite sensitive around 1 kHz to 3 kHz, and is able to hear
sound of smaller intensity level than other frequencies.

Acoustics                                   Page 6                                     Sep 2007
AL Physics/Wave Motion                                                                  K.W. LAM

1.23 Noise Pollution

Noise – unwanted sound, unpleasant to human ear.

Sources of Sound Pollution:

    traffic (e.g. motor vehicles)
    machines (e.g. industry and domestic appliances)
    aircrafts
    construction sites
    greatly amplified music
    noise from people (e.g. talking, activity, mahjong)

Damage to human:

    damage to the ear/hearing
    tiredness and loss of concentration
    sickness and temporary deafness

1.24 Typical Noise Levels in Everyday Life

The noise level is usually measured in dB. The table below shows typical noise levels
corresponding to various situations.

Noise level in                   Sources of noise
dB
Very loud               120            Near roar of airplane (Threshold of pain)
110            Violent hammering of steel plate/ Thunderclap
100            Disco
Loud                   90            Loud motor horn/
Normal traffic noise in Hong Kong
70            Busy office
Moderate                 60            Urban district
50            Normal conversation
40            Inside a running car
Quiet                  30            Suburban street/ Hospital wards
20            Quiet room/ libraries
10            Faintest audible sound
Silent                  0            Threshold of hearing

Acoustics                                       Page 7                                     Sep 2007
AL Physics/Wave Motion                                                           K.W. LAM

1.25 Absorption of Sound and Sound Proofing

 Designing quieter engines and better exhaust systems.
e.g. car engines are mounted on metal brackets via rubber blocks which absorbed
vibration.

 Using sound-insulating materials at homes, e.g. carpets and curtains.

 Fibreglass is a good material to absorb sound waves. It can convert the sound energy
into internal energy.

 Materials containing air cavities (e.g. sponge) of different sizes are also useful to
absorb sound over a range of frequencies. In this case, sound energy is transferred to
resonate the air in the air cavities.

 Buildings besides the highway are usually built in thicker wall. Moreover, double
glazed windows are installed too.

 Barriers are erected alongside the highway to reduce the noise.

 People exposed regularly to noise (e.g. tractor drivers, factory workers, and pneumatic
drill operators) can wear ear protectors to protect their ears.

Notes:
High-pitched noises are more annoying to human ear than low pitched ones.

Related AL Question: [85/IIB/4(a)(b)]

Acoustics                                     Page 8                                Sep 2007
AL Physics/Wave Motion                                                            K.W. LAM

2. Velocity of Sound

2.1 Measurement of the Speed of Sound by Kundt’s Tube

Aim: To produce longitudinal stationary waves in the air inside a closed tube and make
measurement to determine the speed of sound.

Procedure:

1. Set up the apparatus as shown below with the loudspeaker placed touching the open
end of the tube.

2. Place a thin layer of dry chalk powder or cork dust along the length of the tube.

3. Set the signal generator at a high output and slowly increase the frequency from 1
kHz. At a certain frequency the chalk powder whirls around the tube and settles into
heaps.

Mark the positions of the nodes and antinodes of the stationary wave in the above
diagram.

4. Measure the distance of a number of node-to-node (or antinode-to-antinode)
separations and find the wavelength. Record the resonant frequency f and calculate
the speed of sound in the air using the formula v = f.

Acoustics                                   Page 9                                   Sep 2007
AL Physics/Wave Motion                                                        K.W. LAM

Results: (Sample data)

Distance of 5 node-to-node separations = 0.445 m
 wavelength  =

Frequency (of signal generator) f = 2 kHz

Speed of sound in air, v =

Related AL Question: [96/II/2(c)]

2.2 Order of Magnitude of Speed of Sound

Medium                Examples             Velocity of Sound (m s-1)
Solid                Steel                         5960
(  103 m s-1)            Glass                         5640
Copper                         4760
Liquid              Mercury                         1450
(  103 m s-1)            Water                         1437
Gas                   Air                          331
(  102 m s-1)       Carbon dioxide                     259

Acoustics                                     Page 10                               Sep 2007
AL Physics/Wave Motion                                                              K.W. LAM

3. Doppler Effect

Doppler effect is a phenomenon such that there is an

apparent change in the frequency of a wave motion when there is relative motion between
the source and the observer.

Note:
1. Doppler effect occurs with EM waves as well as sound waves.
2. For EM waves, the relative speed of source and observer must be small compare with the
speed of EM waves.

3.1 Change in Observed Frequency

S is the source of waves of frequency f and velocity v and O is the stationary observer.

If S is at rest, in 1 second, f waves are emitted and occupy a distance v.

3.11 Source Moving

(1) Towards a Stationary Observer

 When S is moving towards O with velocity us,
 f waves are compressed into smaller distance (v - us)
 to O, the effect is a decrease of wavelength

v  us
Apparent wavelength, 0 =
f

Acoustics                                    Page 11                                   Sep 2007
AL Physics/Wave Motion                                                        K.W. LAM

 If f0 is the apparent frequency (since wavelength decreases)

wave velocity
f0 =
apparent wavelength
v
=
v  us
f
v
f0 = (         )f
v  us
v
      f0 > f       (since          >1)
v  us
 Apparent frequency > Actual frequency
 Pitch higher

(2) Away from a Stationary Observer

 When S is moving away from O with velocity us,
 f waves are extended to larger distance (v + us)
 to O, the effect is an increase of wavelength

v  us
Apparent wavelength, 0 =
f

 If f0 is the apparent frequency (since wavelength increases),

wave velocity
f0 =
apparent wavelength
v
=
v  us
f
v
f0 = (         )f
v  us
v
      f0 < f       (since          <1)
v  us
 Apparent frequency < Actual frequency
 Pitch lower

Acoustics                                    Page 12                            Sep 2007
AL Physics/Wave Motion                                                            K.W. LAM

Q: The velocity of the moving source should lie in the line joining the source
and the observer. If it makes an angle with the line joining the source
and the observer, what will happen to the apparent frequency as the
source passes the observer?

3.12 Observer Moving

(1) Towards a Stationary Source

 wavelength of wave is unchanged. (Note: It is NOT compressed.)

 velocity of wave relative to O is v + uo

relative vielocity of wave
Apparent frequency, f0 =
wavelength
v  uo
=
v
f
v  uo
f0 = (        )f
v
v  uo
      f0 > f    (since         >1)
v
 Apparent frequency > Actual frequency
 Pitch higher

(2) Away from a Stationary Source

 wavelength of wave is unchanged. (Note: It is NOT compressed.)

 velocity of wave relative to O is v  uo

relative vielocity of wave
Apparent frequency, f0 =
wavelength

Acoustics                                     Page 13                                  Sep 2007
AL Physics/Wave Motion                                                               K.W. LAM

v  uo
=
v
f

v  uo
f0 = (    )f
v
v  uo
       f0 < f   (since         <1)
v
 Apparent frequency < Actual frequency
 Pitch lower

3.13 Source and Observer Moving

Motion of source affects the apparent wavelength of wave.
Motion of observer affects the relative velocity of the wave he/she received.

(1) Approaching each other

 velocity of wave relative to O, v0 = v + uo (due to moving observer)
v  us
 Apparent wavelength, 0 =             (due to moving source)
f
v            v  uo
 Apparent frequency, f0 = 0 =
0           v  us
f
v  uo
f0 = (          )f
v  us
v  uo
        f0 > f    (since          >1)
v  us
 Apparent frequency > Actual frequency
 Pitch higher

(2) Away from each other

 velocity of wave relative to O, v0 = v - uo (due to moving observer)
v  us
 Apparent wavelength, 0 =             (due to moving source)
f
v            v  uo
 Apparent frequency, f0 = 0 =
0            v  us
f
v  uo
f0 = (          )f
v  us
v  uo
        f0 < f    (since          < 1)
v  us

Acoustics                                    Page 14                                   Sep 2007
AL Physics/Wave Motion                                                        K.W. LAM

 Apparent frequency < Actual frequency
 Pitch lower

Summary:

In general,
v  uo
f0 = (          )f
v  us

upper signs  towards each other
lower signs  away from each other

Example

A car moving at 20 ms-1 sounds the horns, which has a frequency of 256 Hz (according to
the driver).
(a) What is the frequency according to a pedestrian
(ii) directly behind?
(b) Suppose the pedestrian blows a whistle at the same frequency. What is the frequency
heard by the driver if he is driving
(i) directly towards, and
(ii) directly away from the pedestrian?
(c) Comment on the frequency changes for part (a) and (b).
(Velocity of sound in air is 330 m s-1.)

Solution

(a) (i)

(ii)

(b) (i)

(ii)

(c) The frequency changes are ________________________ for parts (a) and (b).

Acoustics                                 Page 15                                Sep 2007
AL Physics/Wave Motion                                                                K.W. LAM

3.2 Real Life Examples

3.21 Police Car and Ambulance

Pitch of the note from the siren of a fast-travelling police car or ambulance seems
higher as it is approaching a stationary observer. On the contrary, the pitch drops
suddenly as it passes the stationary observer.

 A radar signal of speed c and frequency f, is sent towards a receding car of speed
u. The moving car receives the radar signal at a lower frequency f1 since the
observer is receding.

cu
f1 =    (       )f
c

 Now the car acts as a moving source of the reflected waves. The radar receives the
reflected signal at an even lower frequency f2.

c                c     cu            cu
f2 =    (       ) f1 =   (       )(     )f =   (       )f
cu              cu     c             cu

So the difference in frequency between f2 (reflected wave by car) and f (initial
f = f 2 – f
cu
= (         )f  f
cu
 2u
=         f
c
2u
i.e., there is a decrease of frequency by      f
c

 By comparing f and f2, the speed of the car u can be computed and see whether it
exceeds the legal limit.

Acoustics                                        Page 16                                 Sep 2007
AL Physics/Wave Motion                                                              K.W. LAM

Example

A police speed check operates by reflecting radio waves of frequency 100 MHz from
passing cars. It is found that the frequency after reception has decreased by 18 Hz.
Should a speeding ticket be issued if the speed limit is 70 km h-1?

Solution

Note:
If the radar signal is sent to an approaching car, the total difference in frequency
2u
between incident and reflected wave is, similarly, f =    f , i.e., there is an increase
c
2u
of frequency by      f.
c

Related AL Questions: [93/IIA/2(c)] [97/IA/2(c)(ii)] [99/IB/6(b)(i)]

3.21 Galaxy Red Shift

The wavelengths due to an identifiable element in a star’s spectrum (obtained by a
________________) are longer than the actual wavelengths, i.e. towards the red end
of the spectrum. This phenomenon is known as ‘Red Shift’.

The change of wavelength and frequency means that there may be relative motion
between the light source (the star) and the observer (on the earth). It indicates the
recession of the star from the earth.

Acoustics                                   Page 17                                     Sep 2007
AL Physics/Wave Motion                                                           K.W. LAM

The same results will be obtained if light is measured from galaxies. This implies that
all galaxies are receding from the earth at a very high speed (up to c/3). In other
words, the universe is ________________.

Acoustics                                  Page 18                                   Sep 2007

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