Sound
Sound is a longitudinal wave that requires a medium for propagation. We usually think
of sound waves in air; however, sound waves can also exist in a liquid such as water or in
a solid. The waves corresponds to regions of compression and expansion in which the
density and pressure of the wave is above or below the average in the medium. These
regions propagate with a speed that depends on the density and elasticity of the medium.
Specifically, the speed of sound is given by
B
v

,
P
is the bulk modulus of the medium and  is the density. This
V / V
expression is analogous to that for the speed of a wave on a string, which is
where B  
v
F

,
where F is the tension and  is the mass per unit length.
For a gas, it can be shown that the speed of sound is
v
RT
M
,
where T is the Kelvin temperature and M the molecular mass.  is the ratio of the specific
heat measured at constant pressure to the specific heat measured at constant volume and
is a property of the gas.
Thus, from the above expression the speed of sound in helium is higher than in air. The
speed of sound also increases with temperature. For room temperature, the speed of
sound in air is about 345 m/s.
Energy carried by sound waves
The energy in a sound wave can be characterized by its intensity, which is the power
transmitted through a surface per unit surface area:
I
power P

area
A
(units = watts/m2)
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A person with a good ear can detect sound with intensity as low as 10-12 W/m2 at a
frequency of about 1000 Hz. Normal conversation corresponds to an intensity of about
10-6 W/m2. A sound with intensity of about 1 W/m2 would be painful to the ear because
it is so intense and could do permanent damage to the ear.
An alternative way of specifying the intensity is to use the decibel scale. The intensity
level in decibels is given as
 I 
 ,
I
 0
  10 log 
where I0 =10-12 W/m2.
Example:
What is the intensity level in decibels of a sound with intensity I = 5 x 10-6 W/m2?
 5 x10  6 
  10 log( 5 x10 6 )  70dB

12
 10



  10 log 
If I = 10-12 W/m2, then  = 10 log(1) = 0 dB (threshold of hearing)
If I = 1 W/m2, then  = 10 log(1/10-12) = 10 log(1012) = 120 dB (pain level)
Example:
What is the intensity in W/m2 corresponding to a decibel level of 55 dB?
I  I 0 10  / 10  10 12 W / m 2 10 55 / 10  10 1210 5.5 W / m 2  10 6.5 W / m 2
 3.16 x10  7 W / m 2
Intensity variation from a point source of sound:
A very small source of sound (a point source) will radiate uniformly in all directions. All
of the power, P, must pass through a sphere of radius r and area A = 4r2. Thus, the
intensity at a distance r from the source is
I
P
P

A 4 r 2
2
Example:
The intensity level of a point source of sound is 100 dB at a distance of 10 m from the
source. What is the intensity level at a distance of 20 m?
 P / 4 r 2 
 I2 
I 
I 
2 
  10 log  1   10 log  2   10 log 
2 

I
I
I
 1
 0
 0
 P / 4 r1 
 2  1  10 log 
r2 
 10 2 
  10 log  1   6.0 dB
 10 log  1   10 log 
 20 2 
r 2 
4


 2 
 2  1  6 dB  100 dB  6 dB  94 dB
Each time the distance from the source is doubled, the intensity level is reduced by 6 dB.
Doppler Effect
The frequency of a wave that is detected by an observer depends on the relative motion of
the source and the observer. If the source and observer are moving toward each other,
then the observer detects an increase in frequency compared to the case when the source
and observer are at rest. If the source and observer are moving away from each other,
then the observer detects a decrease in frequency. This phenomenon is called the
Doppler Effect. The detected frequency is given by
 v  vo 

f '  f 
 v  vs 
In this equation f is the frequency of the source as measured with no relative motion, v is
the speed of sound, vo is the speed of the observer, and vs is the speed of the source. If
the observer is moving toward the source, then vo is positive. If the observer is moving
away from the source, then vo is negative. Likewise, vs is positive if the source is moving
toward the observer and negative is the source is moving away from the observer.
Example:
A train is traveling at 50 m/s with the whistle blowing at a frequency of 1000 Hz as heard
by the passengers in the train. What frequency does a person standing by the tracks hear
as the train approaches and as the train leaves?
Approaching:
 345  0 
f '  1000 Hz 
  1169 Hz
 345  50 
3
Leaving:
 345  0 
f '  1000 Hz 
  873 Hz
 345  50 
Example:
A police car traveling at 40 m/s chases a motorist traveling at 30 m/s. The police siren is
turned on and is heard by the policeman at a frequency of 500 Hz. What frequency does
the motorist hear?
 345  30 
f '  500 Hz 
  516 Hz
 345  40 
Breaking the sound barrier:
When an object travels faster than the speed of sound, then it creates a cone-shaped shock
wave.

vt
vst
The angle that the shock wave makes with the direction of travel of the source is given by
sin  
v
vs
Standing waves on a string
Waves can interfere either constructively or destructively, depending on their relative
phase when they combine. If waves are generated on a string that is clamped at both ends,
then the waves will be reflected at the clamped ends. The resulting wave will consist of a
sum of waves going left and right. If the length of the string is an integral number of
half-wavelengths (L = n /2), then ‘standing waves’ will be formed. These consist of
regions where the right and left-going waves interfere constructively (antinodes) and
interfere destructively (nodes).
4
/2
node
antinode
Allowed standing wave frequencies:

L  n , n  1, 2, 3, ...
2
2L

n
v
v
v
f  
n
 2L / n
2L
F
v
.

fn 
n
2L
F

, n  1, 2, 3, ...
n  1 : f1 
1
2L
n  2 : f 2  2 f1
n  3 : f 3  3 f1
F

1st harmonic (or fundamental )
2nd harmonic (or 1st overtone)
3rd harmonic (or 2nd overtone)
etc.
Example:
A string 0.8 m long that is clamped at both ends has a tension of 100 N and a linear mass
density of 0.02 kg/m. What are the three lowest allowed standing wave frequencies?
fn 
n F
n
100

 n 44.2 Hz
2 L  2(0.8) 0.02
f1  44.2 Hz , f 2  2 f1  88.4 Hz , f 3  3 f1  132.6 Hz
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Standing waves in air columns
Sound waves in an air column can be reflected from either a closed end or an open end.
If the column has an appropriate length relative to the wavelength of the sound, then
standing waves can be formed in the column. A closed end will be a node since the air
movement at the end will be restricted. An open end will be an antinode since the air at
the open end is unrestricted and able to undergo a maximum displacement.
Open-open column:
1 = 2L, f1 = v/1 = v/2L
2 = L, f2 = v/2 = v/L
1 = 2L/3, f3 = v/3 = 3v/2L
General result for open-open: f n  n
v
, n  1, 2, 3, ...
2L
Open-closed column:
1 = 4L, f1 = v/1 = v/4L
2 =4L/3, f2 = v/2 = 3v/4L
1 = 4L/5, f3 = v/3 = 5v/4L
General result for open-closed: f n  n
v
, n  1, 3, 5, ...
4L
So, an open-open pipe behave somewhat like a string clamped at both ends. All
harmonics (n = 1, 2, 3, …) are allowed. For an open-closed pipe, however, only odd
harmonics are allowed (n = 1, 3, 5, …). Another difference between the open-open and
open-closed pipes is that the fundamental frequencies differ by a factor of two for the
same pipe length.
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Example:
What are the three lowest standing wave frequencies for an open-open pipe and an openclosed pipe, each of length 0.4 m?
Open-open:
v
345 m / s
n
 n 431.25 Hz
2L
2(0.4m)
f1  431.3 Hz , f 2  2 f1  862.5 Hz , f 3  3 f1  1293.8 Hz
fn  n
Open-closed:
v
345 m / s
n
 n 215.6 Hz
4L
4(0.4m)
f1  215.6 Hz , f 3  3 f1  646.9 Hz , f 5  5 f1  1078.1 Hz
fn  n
Beats
Two sinusoidal sound waves of different frequencies when combined interfere
constructively to produce ‘beats’. The ‘beat’ frequency is just the difference in frequency
of the two sound waves.
f1
+
f2
||
fbeat = f2 – f1
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Quality of Sound
Most sounds are not simple sinusoidal waveforms with a single fixed frequency and
amplitude that last indefinitely. Typical sounds are transient in nature and consist of a
combination of sine wave tones (harmonics). The ‘quality’ of the sound is determined by
the relative intensity of the different harmonics that make up the sound and the transitory
nature of the sound (the build up, duration, and decay of the sound). The quality of the
sound produced by an open-open organ pipe and an open-closed pipe will differ since the
open-open pipe can resonate with any combination of harmonics, whereas the openclosed pipe can resonate only with a combination of odd harmonics. The ‘pitch’ that we
associate with a sound is typically related to the first harmonic, even if higher order
harmonics also exist.
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