Announcements 10/10/11
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Prayer
Exams graded, see email
In HW 17-5b: be very careful to track the correct
peak when plotting it for t = 0.1 s and t = 0.5 s,
and when calculating the velocity of the peak.
Dispersion Review
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Any wave that isn’t 100% sinusoidal contains more
than one frequencies.
To localize a wave in space or time, you need lots of
frequencies--spatial (k values) or angular (w values),
respectively. Really an infinite number of frequencies
spaced infinitely closely together.
A dispersive medium: velocity is different for different
frequencies.
Two Different Velocities
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What happens if a wave pulse is sent through a
dispersive medium? Nondispersive?
Dispersive wave example:
a. f(x,t) = cos(x-4t) + cos(2 (x-5t))
– What is “v”?
– What is v for w=4? What is v for w=10?
What does that wave look like as time progresses?
(next slide)
Mathematica
0.1 seconds
0.7 seconds
What if the two
velocities had
been the same?
1.3 seconds
Time Evolution of Dispersive Pulse
Credit: Dr. Durfee
Power spectrum
Peak
moves at
about 13
m/s (on
my office
computer)
Wave moving in time
Note:
frequencies
are infinitely
close together
How much energy is
contained in each
frequency component
Phase and Group Velocity
Credit: Dr. Durfee
Window is moving
along with the
peak of the pulse
vp 
w
k
 velocity of "wiggles"
Can be different for each
frequency component
that makes up the wave
12.5 m/s, for dominant component
dw
vg 
dk
 velocity of "envelope"
evaluatedat kave
13 m/s
(peak)
A property of the wave
as a whole
From Wikipedia
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Example where vphase > vgroup
http://en.wikipedia.org/wiki/Group_velocity
One of my contributions to Wikipedia
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Example where vphase is negative!
http://en.wikipedia.org/wiki/Group_velocity
Thought question
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A wave at frequency ω traveling from a string to a rope.
At the junction, 80% of the power is reflected. How
much power would be reflected if the wave was going
from the rope to the string instead?
a. Much less than 80%
b. A little less than 80%
c. About 80%
d. More than 80%
e. It depends on the color of the rope.
AR v2  v1
r
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AI v1  v2
AT
2v2
t
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AI v1  v2
R  r2
T  1 R
Demo
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Reflection at a boundary. Measure v1 and v2.
v2  v1
r
v1  v2
2v2
t
v1  v2
Reading Quiz
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Sound waves are typically fastest in:
a. solids
b. liquids
c. gases
Sound Waves
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What type of wave? What is waving?
Demo: Sound in a vacuum
Demo: tuning fork
Demo: Singing rod
Sinusoidal?
a. Demo: musical disk
vsound  343 ms
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T
293K
Speed of sound
Speed of sound…
a. in gases: ~300-1200 m/s
b. in liquids: ~1000-1900 m/s
c. in solids: ~2000-6000 m/s
v = sqrt(B/r)
compare to v = sqrt(T/m)
Speed of sound in air
a. 343 m/s for air at 20C
b. Dependence on Temperature (eqn in book
and also given on exam)
vsound  343 ms
T
293K
Intensity
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Intensity: power/area
a. “Spherical waves”
b. Non-spherical waves?
Question: you measure the sound intensity
produced by a spherically-emitting speaker to
be 10 W/m2 at a distance of 2 meters. What
will be the intensity at 8 meters away?
Question: What is the total sound power
(watts) being produced by the speaker?
Reading Quiz
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How do we calculate the sound level in
decibels?
a. β = 10 log( I / Io )
b. β = 10 ( I / Io )
c. β = 10 ( I - Io )
d. β = 10 e( I / Io )
e. β = e10 ( I / Io )
add 10 to b  10 to I
Decibels
Threshold of hearing
0 dB
10-12 W/m2
Whisper
30 dB
10-9 W/m2
Vacuum cleaner
70 dB
10-5 W/m2
Rock Concert
120 dB
1 W/m2
Nearby jet airplane
150 dB
1000 W/m2
Logarithm Review
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Log10(x) is the inverse of 10y
→ if x = 10y then y = log10(x)
a. I.e. “10 to the what equals 22?”
answer: 1.3424
calculator: log10(22)
Review of “Laws of Logs”:
– 1. log(ab) = log(a) + log(b)
– 2. log(an) = n log(a)
log10(100) = ? Translation: 10 to what equals 100?
ln(100) = ?
(“ln” = loge = log2.71828…)
Translation: e to what number =100? (4.605…)
Ambiguity: “log(100)”…could be either log10 or ln
Question: log10(1,000,000) = ?
Question: If log(3) = 0.477, what is log(300)?