11.1 Propagation of Sound Waves
> Sound exhibits properties of waves like reflection, refraction, diffraction and interference. It does not
exhibit polarization. This suggests sound is a longitudinal wave.
> Sound wave is produced by a vibrating object and it needs a material medium for transmission. The
vibrating object causes the particles of the transmitting medium to oscillate and carries the energy from
the source to the receiver.
> When sound is transmitted through the air, a series of compressions (high pressure regions) and
rarefactions (lower pressure regions) moves away from the source at the speed of sound. However the air
particles themselves just vibrate back and forth about their equilibrium position. Figure 2 shows the
variation of pressure with distance for the sound wave. Notice that pressure (density) is maximum at
compression whereas at the centre of compression, the displacement is zero . The pressure variation is
said to have a phase leads of
rad compared with the displacement variation. The wave equation in
terms of displacement and pressure can thus he written as:
y = yo sin (wt – kx)
p=pA+po sin(wt-kx+
)
or excess pressure or change in pressure from the normal air pressure is given by:
p – pA = po sin (wt – kx +
)
11.2 Sources of sound
> Stretched strings
When a stretched string is plucked, bowed or struck, two progressive waves travel from the point in
opposite directions and are reflected from the fixed ends. A stationary wave pattern is formed for waves
whose wavelengths fit into the length of the string.
Note
(a)The string can sound all the harmonics f, 2f, 3f, etc.
(b)The string always has odd numbers of (nodes + antinodes)
The speed v of the transverse wave along a stretched string of mass per unit length la an tension T is
given by:
Resonance occurs for those waves whose wavelengths are correctly matched to tt length of the string.
The wavelength of the stationary wave formed,
> Air columns
Stationary longitudinal waves in air columns in a pipe are the source of sound in wit instruments. To set
the air into vibration, a disturbance is created at one end (open) of tl pipe, the other end can be open or
closed.
Open Pipe
Note (a)Fundamental frequency of open pipe is twice the fundamental frequency of closed pipe of the
same length.
(b)The open pipe can sound all the harmonics f, 2f, 3f, etc.
(c) In practice the antinode of a stationary wave at the open end of a pipe occurs at a distance c (end
correction) beyond the end.
> Resonance tube (Use to determine l or v)
The effective length of the air column is greater than the length of the pipe because of end correction.
…………..(1)
…………(2)
(2) – (1)
> Vibrating membranes
A piece of stretched membrane can vibrate with modes as shown in Figure below. These vibrations are
two dimensional vibrations.
Example 1
Neglecting end effect, find the length of an open organ pipe that emits a fundamental note of frequency
256 Hz. (Take the speed of sound in air to be 330 m s-1)
Solution
Example 2
An open pipe of length 30 cm is sounding its second overtone. The note emitted is having wavelength of
22 cm. What is the end-correction of the pipe?
Solution
11.3 Intensity of Sound
> Intensity of sound is the rate of flow of energy per unit area perpendicular to the direction of travel of
the sound. Intensity is measured in W m -2.
> At a distance r from a source with power P, all the energy is spread out over a sphere of radius r, so the
intensity is
The level of intensity L is defined by
where I0 = 10-12 W m-2 is known as the threshold of hearing which is the lowest intensity at a given
frequency that can be heard.
The unit for the level of intensity is decibel (dB).
Example 3
When a student stands at a distance 20 m from a sound source, the level of intensity he receive is 65 dB.
Assuming that the sound source is a point source, calculate:
(a) The intensity the student receives.
(b) The power of the sound source.
Solution
11.4 Beat
> When two notes of slightly different frequencies but similar amplitudes are sound( together, the
loudness increases and decreases periodically and beats are said to be hear Displacement
> Beat frequency
Suppose the Beat period (i.e time between two successive maxima) is T and the frquencies of the two
note, of slightly different frequencies are f1 and f2 .
Number of cycles of frequency f1 in time T = f1T
Number of cycles of frequency f2 in time T = f2T
f1T-f2T= 1
\ beat frequency = f =
= f1 – f2
Example 4
When a tuning fork of unknown frequency f is sounded together with another tuning fork of frequency
1000 Hz, 4 beats are heard in one second. When a small piece of plasticine is attached to the prong of
the tuning fork with unknown frequency and the two tuning forks are sounded together again, 2 beats are
heard in one second. What is the unknown frequency f?
Solution
When 4 beats are heard, the unknown frequency is either 996 Hz or 1004 Hz
When plasticine is attached to the tuning fork, 2 beats are heard, so the frequency is either 1998 Hz or
1002 Hz.
However when the platicine is attached to the tuning fork, it decreases the original frequency I (i.e. 1004
Hz -) 1002 Hz), so the unknown frequency f is 1004 Hz.
Example 5
Two sound waves have frequencies 960 Hz and 964 Hz respectively. The intensity of each wave at a
particular point is 6.5 x 10-9 W m-2.
(a) What is the beat frequency that can be heard at that point?
(b) Determine the level of the maximum intensity of the beat that could be heard at that point.
Solution
11.5 Doppler Effect
> The Doppler Effect is the apparent change in frequency of a wave due to the relative motion between
the source and observer.
Let the observer be denoted by O and the source by S. When O and S are stationary in still air, v = fl
where v is the velocity of the wave, f the frequency of the source and l the wavelength.
> S stationary, O moving towards S at velocity u0
The wavelength l is unchanged, but the velocity of the wave relative to O is increased from v to (v + n o).
The observed frequency is increased from f to f’
where
\
> S stationary, O moving away from S at velocity u 0
> O stationary, S moving towards O at velocity u s
The velocity v of the wave is unchanged but the wavelength will become shorter. In one second, f waves
has been emitted in a distance of (v-us)
therefore
=
> O stationary, S moving away from O at velocity Us
the velocity v of the wave is unchanged but the wavelength will become longer. In one second , f waves
has been emitted in a distance of (v+us).
therefore
=
> In general,
where the upper signs correspond to moving towards and the lower signs correspond to moving away.
Example 6
A car P moving with velocity 30 m s-1 sounds a note of 1000 Hz from its horn. What is the apparent
frequency heard by the driver in a car Q moving behind in the same direction with velocity 20 m s -1.
(Taking velocity of sound in air as 300 m s-1)
Example 7
A boy is walking away from, a wall at a speed of 2.0 ms -1 in a direction at right angle to the wall. As he
walks. he blows a whistle steadily. An observer towards whom, the boy is walking hears 6.0 beats per
second. If the speed of the sound is 340 ms-1 what is the frequency of the whistle ?