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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
        bad tempered and nervous
        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
                                 80            Loud radio music
                                 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
       (i) directly ahead, and
       (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.




3.21 Radar Speed Traps

         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
             incident wave by radar),
                                  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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