Announcements 2/16/11
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Prayer
Exams… hopefully graded by Friday
Extra time on Lab 3: now due Tues Feb 22.
Monday Feb 21 is President’s Day holiday.
Tues Feb 22 is a virtual Monday
Remember in HW 17-5b to 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.
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
Dispersion Summary
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A dispersive medium: velocity is different for
different frequencies
Any wave that isn’t 100% sinusoidal contains
multiple frequencies.
To localize a wave in space or time, you need lots
of frequencies… really an infinite number of
frequencies spaced infinitely closely together.
Two Different Velocities
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What happens if a wave pulse is sent through a
dispersive medium? Nondispersive?
Dispersive wave example:
a. s(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
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
On Wikipedia
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Example where vphase > vgroup
http://en.wikipedia.org/wiki/Group_velocity
Not yet on Wikipedia
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Example where vphase is negative!
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
Speed of sound
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Speed of sound…
a. in gases: ~300-1200 m/s (343 m/s for air at
20C)
b. in liquids: ~1000-1900 m/s
c. in solids: ~2000-6000 m/s
General form: v = sqrt(xxx/yyy)
a. What are “xxx” and “yyy”?
Speed of sound in air
a. Dependence on Temperature (eqn in book)
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
Thought Question
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A 3 dB increase in intensity is just about a
factor of 2. How many dB represents a
factor of 4 increase in intensity?
a. 2
b. 4
c. 6
d. 8
e. 9
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 (log(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 number equals 100?
ln(100) = ?
(“ln” = loge = log2.71828…)
Translation: e to what number =100? (4.605…)
If the problem just says log(100)…could be either log10 or ln
Question: log10(1,000,000) = ?
Question: If log(3) = 0.477, what is log(300)?
Power and Intensity Scales
Power or Intensity
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dB
dBW
dBm
β = 10 log(I/I0)
β = 10 log(P/P0)
β = 10 log(P/P0)
What if you need to solve for I?
I0 = 10-12 W/m2
P0 = 1 W
P0 = 1 mW