Sound Waves
•
A sound wave is a longitudinal wave, like on a slinky
λ
Displacement
of molecules
from equilibrium
2
1.5
s (m)
1
0.5
0
-0.5 0
1
2
3
4
5
6
-1
-1.5
-2
2
x (m)
1.5
∆ p (Pa)
Pressure
offset
1
0.5
0
-0.5 0
-1
1
2
3
4
5
6
-1.5
-2
x (m)
Phy1222 - Spring 2003
1
Sound Waves
•
There is a particular relation between the displacement and the
pressure:
s = s m cos(kx − ωt )
∆p = ∆pm sin( kx − ωt )
∆p m = ( vρω ) s m
Pressure amplitude =
(wave velocity)(density of medium)(angular frequency)(displacement amplitude)
•
Example: If a 440 Hz sound wave in air, travelling at 343m/s, has a
pressure amplitude of 0.15 Pa, how far are the air molecules moving
back and forth?
Phy1222 - Spring 2003
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1
Power and intensity
•
•
Sound waves spread out from the source in 3-dimensions
If the source is point-like, then the wave fronts are spherical
Ps
r
•
•
dA
The energy of sound per unit time is the power of the source (P s)
The power per unit area at some distance r from the source is the
intensity:
∆P
P
“inverse-square law”
I=
∆A
=
s
4πr 2
(isotropic source)
Intensity = (power of source)/(4π(distance from source)2)
Phy1222 - Spring 2003
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Power and Intensity
•
The intensity can also be expressed in terms of the properties of the
sound wave itself:
(note the close resemblance
1
to the power of a wave on a string:
2 2
I=
2
ρvω sm
P=
1
µvω 2 ym2 )
2
Intensity=
(1/2)(density of medium)(wave velocity)(ang. Freq.)2(displacement amplitude)2
•
Example: For a 440 Hz sound wave with a displacement amplitude of
13.2 microns (in 20o air)
– what is the intensity?
– How much power is available to your ear drum (area ~1cm2)?
– If this is 10m from the point-like source of the sound, what is the
total power of the source?
Phy1222 - Spring 2003
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2
Decibel Scale
•
Reference intensity for sound (around lower limit for human hearing):
I 0 = 10 −12 W/m 2
•
Logarithmic scale:
 I
β = (10dB ) log 10  
 I0 
Start from 0 at I0,
and add 10 for each
factor of 10
•
For example, 50 dB is 10 times more intense than 40 dB, and 100
times more intense than 30 dB.
•
•
Pain threshold: around 120 dB.
Example: For the 440 Hz sound considered in the previous example
(I=0.274 W/m2 ),
– What is the intensity in dB?
– How close could you come to the source without pain?
Phy1222 - Spring 2003
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Human Hearing
•
Huge ranges of sensitivity:
– Intensity
• min ~10-12 W/m2 (0 dB)
• max ~1 W/m2 (120 dB)
– Frequency
• min ~20 Hz
• max ~20 kHz (decreases with age and exposure to loud noises)
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3
Speed of Sound
•
•
Speed of sound in 20oC air: about 343 m/s (you measured this in the
lab with the air-track experiments with the ultrasonic sensors).
Speed of sound in any medium:
“springiness”
v=
B
ρ
“inertia”
Speed of sound propagation
= square root((bulk modulus of medium)/(density of medium))
•
The bulk modulus, B, is an elastic property of the material (like the
Young’s modulus, see sec 13-6). It measures the incompressibility of
the material.
∆V
∆V
p=B
V
p
V
Pressure =
(Bulk modulus)(change in volume)/(volume)
Phy1222 - Spring 2003
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Speed of Sound
•
•
Careful: sound is not necessarily slower in a more dense material,
because often more dense materials are also more incompressible.
Sound tends to be fastest in solids, and slowest in gasses
– 6000 m/s in steel
– 1482 m/s in water
– 343 m/s in air
•
Example: You see a lightning flash, and hear the thunder about 3
seconds later. How far away did the lightning strike?
Phy1222 - Spring 2003
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4
Wind Instruments
•
•
•
Same idea as string instruments: particular frequency will “resonate”
as a standing wave in the structure and produce sound of that
frequency.
For wind instruments, the resonant material is air (usually in some sort
of tube), and the wave speed is always the speed of sound.
No “tension” that you can tune. Only the length of the tube, and the
choice of “open” or “closed” ends.
Open end:
displacement anitnode,
pressure node
Closed end:
displacement node,
pressure antinode
(show here: displacement)
closed-open (shown here):
λ=4L,4L/3,4L/5,…4L/(2n-1)
open-open (not shown)
λ=2L,2L/3,L,…4L/n
Phy1222 - Spring 2003
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Wind instuments
•
Example: A 1-meter tall column is partially filled with water, and 440
Hz tuning fork is vibrating at the open end. For what depths of water
between 0 cm and 100 cm will there be a resonance?
Phy1222 - Spring 2003
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5
Interference (same wavelength)
•
Two sources, same frequency, in phase: Interference effect is
determined by the difference in path length expressed in
wavelengths:
d =λ
Constructive
interference
d =λ/2
Destructive
interference
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Interference (same wavelength)
•
Example: Two speakers driven by the same 500 Hz oscillator are
separated by 4.0 meters. As you walk from one to the other,
– Where does the first sound reach a minimum?
– How many minima do encounter?
4.0 m
2.0 m
– If you walk along a parallel path 2.0 m to the side, where to you
find the first minimum?
Minima: d=λ(n+1/2)
Maxima: d=λ(n)
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6
Interference: different frequencies (beats)
•
Superposition of two waves of slightly different frequencies gives a
sort of bean-pod shaped wave form:
2
1
0
y (m)
-1 0
5
Beat period
10
15
20
y1
y2
y1+y2
-2
-3
-4
-5
-6
-7
Tone period
x (m)
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Beats
•
Mathematically:
s (t ) = 2 sm cos
( (ω
1
2
1
) ( (ω
− ω 2 )t cos
The volume is
modulated with
the beat frequency:
fbeat = f1 − f 2
1
2
1
+ ω 2 )t
)
The tone has
the average frequency:
ftone =
1
2
( f1 + f 2 )
(useful in tuning musical instruments)
•
Example: combining a 441 Hz sound with a 439 Hz sound will
produce something that sounds like concert A (440 Hz), but “beats”
two times per second.
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7
Doppler Shift
•
Moving source (speaker, siren, etc.) or moving detector (ear,
microphone, …) can shift the frequency:
Lower frequency
for receding source
 v ± vd
f ′ = f 
 v m vs
vs



Higher frequency
for approaching source
v
Detected frequency = source frequency (sound speed)+/-(detector speed)
(sound speed)-/+(source speed)
Phy1222 - Spring 2003
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Doppler Shift
•
Example: You drive toward the wall of a canyon at 25 m/s and honk
your horn (f=256 Hz)
– What is the frequency heard by someone standing at the canyon
wall?
– What is the frequency you hear of the echo off the canyon wall?
– When the reflected sound interferes with the original sound, what
frequency of beats do you “hear”?
25m/s
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8
Shock Waves
•
For a source velocity greater than the speed of sound, the wavefronts
build up into a conical shock wave:
v s∆t
vs
θ
v∆t
v
sin θ =
v
vs
Sine of Mach cone angle =
(sound speed)/(source speed)
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Doppler Shift for light
•
Shift depends only on relative velocity u of source and detector:
u
∆λ = λ  
 c
For u<<c
Change in wavelength =
(wavelength of source)(relative speed)/(speed of light)
•
different from sound formula because relativistic motion compresses
distances and expands time (Physics 4)
•
algebraic sign:
– u positive: motion away, wavelength higher, frequency lower:
“red shift”
– u negative: motion toward, wavelength lower, frequency higher:
“blue shift”
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