Chapter 16 - Sound Waves
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Speed of Sound
Sound Characteristics
Intensity
Instruments: Strings and Pipes
2 Dimensional Interference
Beats
Doppler Effect
Sonic Boom and shock waves
Sound Speed
v
Air
Bulk modulus
B

density

Sea Water
Bulk Modulus 1.4(1.01 x 105) Pa 2.28 x 109 Pa
Density
1.21 kg/m3
1026 kg/m3
Speed
343 m/s
1500 m/s
Variation with Temperature:
Air
Seawater
m
v   331  0.60T 
s
m
v  1449.05  4.57T  .0521T  .00023T 
s
2
3
Pitch is frequency
Audible
20 Hz – 20000 Hz
Infrasonic
< 20 Hz
Ultrasonic
>20000 Hz
Middle C on the piano has a frequency of 262 Hz.
What is the wavelength (in air)?
1.3 m
Intensity of sound
• Loudness – intensity of the wave. Energy
transported by a wave per unit time across a unit
area perpendicular to the energy flow.
Source
Intensity (W/m2)
Sound Level
Jet Plane
100
140
Pain Threshold
1
120
Siren
1x10-2
100
Busy Traffic
1x10-5
70
Conversation
3x10-6
65
Whisper
1x10-10
20
Rustle of leaves
1x10-11
10
Hearing Threshold 1x10-12
1
Sound Level - Decibel
 I
  dB  10log  
 I0 
I0  1x10
12
W
m2
Stringed instruments
v nv
fn 

 n 2L
n=1,2,3,4,....
Question 1
• A steel wire in a piano has a length of 0.9 m and a
mass of 5.4 g. To what tension must this wire be
stretched so that its fundamental vibration
corresponds to middle C: i.e., the vibration possess
a frequency 261.6.
Wind instruments – Double open ended pipes
Frequencies are identical
to waves on a string
fn 
v nv

 n 2L
n=1,2,3,4,....
Wind instruments – Single open ended pipes
Only odd harmonics are present
v nv
fn 

 n 4L
n=1,3,5,7,....
Question 2 – Pepsi Bottle
• What is the fundamental frequency of a pepsi
bottle 32 cm tall when you blow over it. Assume
the speed of sound in air is 343 m/s.
• 5 cm of water are added to the bottle. What is the
new resonant frequency.
32 cm
Waves on the surface of a liquid
Two dimensional wave reflection
i  r
Interference in Space
When the path lengths from source to receiver
differ by /2 destructive interference results.
Interference in Time - Beats
• Two sounds of different frequency:
D1  x  0   D m sin  k1  0   1t   D m sin  2f1t 
D2  x  0  Dm sin  2f 2 t 
• Superposition:
D  D1  D2  Dm sin 2f1t   Dm sin 2f 2 t 
Interference in Time - Beats
D  D1  D2  Dm sin 2f1t   Dm sin 2f 2 t 
• Trig identity again:
sin 1  sin 2  2sin
1  2
 
cos 1 2
2
2
  f1  f 2     f1  f 2  
D  2Dm cos  2 
 t  sin 2 
 t
  2     2  
Amplitude varies in time at a frequency
equal to the difference in the two frequencies
Beat Frequency
f beat  f1  f 2
Beats
f beat  f1  f 2
Doppler Effect
Doppler Effect – 4 cases
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Source moving toward receiver
Source moving away from receiver
Receiver (observer) moving towards source
Receiver (observer) moving away from source.
Source moving case
    vs T
Away:

 1
f  f 
 1  vs
v







v
T
Towards:

 vs 
    v s    1  
v
v


 1
v
v
f  
f

 v 
 1  vs
 1  s 
v

v






Receiver (observer) moving case
Towards:
f 
v  vO
 vO 
 f 1 


v


Away:
v  vO
 vO 
f 
 f 1 


v 

Source and receiver moving
vO

 1 v
f  f 
v
 1 s
v



 v  vO 
f

 v vS 


• Numerator – Receiver (observer)
– Toward +
– Away –
• Denominator – Source
– Toward –
– Away +
Doppler Example
• Intelligence tells you that a particular piece of
machinery in the engine room of a Soviet Victor III
submarine emits a frequency of 320 Hz. Your sonar
operator hears the machinery but reports the frequency
is 325 Hz. Assume you have slowed to a negligible
speed in order to better hear the Russian.
– Is the VIII coming toward you or moving away from you?
– Assuming the Victor is either moving directly toward or
away from you, what is his speed in m/s?
Shock waves and the sonic boom
vsound t vsound 1
sin  


vobject t vobject m
Sometimes you hear 2 booms