announcements 10/5/12
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Announcements 10/5/12. Prayer Handout – Adding together two cosine waves Colloquium: Did you notice “Fourier transforms”? I just got the exams from the Testing Center, TA & I will work on grading them today & this weekend. Non Sequitur. From warmup. Extra time on? - PowerPoint PPT PresentationTRANSCRIPT
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Announcements 10/5/12 Prayer Handout – Adding together two cosine waves Colloquium: Did you notice “Fourier
transforms”? I just got the exams from the Testing Center,
TA & I will work on grading them today & this weekend.
Non Sequitur
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From warmup
Extra time on?a. how exactly can an amplitude absorb a
complex number when it itself is not complex? Is it related to the way you lump a constant into +C after taking an integral?
Other comments?a. (none in particular)
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Adding together two cosine waves
In short: “The amplitude and phase of the answer were completely determined in the step where we added the amplitudes & phases of the original two cosine waves, as vectors.”
Don’t worry about writing each step completely. a.Don’t write “Real( )”b.Don’t write “e i (3x)”
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HW 16.5: Solving Newton’s 2nd Law
Simple Harmonic Oscillator (ex.: Newton 2nd Law for mass on spring)
Guess a solution like
what it means, really:
there’s an understood “Real{ … }”
2
2
d x kx
mdt
( ) i tx t Ae
( ) cos( )x t A t
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Complex numbers & traveling waves
Traveling wave: A cos(kx – t + )
Write as:
Often:
…or – where = “A-tilde” = a complex number
the amplitude of which represents the amplitude of the wave
the phase of which represents the phase of the wave
– often the tilde is even left off
( ) i kx tf t Ae ( ) i kx tif t Ae e
( ) i kx tf t Ae A
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Clicker question: Which of these are the same?
(1) A cos(kx – t)(2) A cos(kx + t)(3) A cos(–kx – t)
a. (1) and (2)b. (1) and (3)c. (2) and (3)d. (1), (2), and (3)
Which should we use for a left-moving wave: (2) or (3)?
a. Convention: Use #3, Aei(-kx-t)
b. Reasons: – (1) All terms will then have same e-it factor. – (2) Whether you have kx then indicates the direction
the wave is traveling.c. “Wavevector”
ˆk k i
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From warmup What was wrong with the first solution that was
tried in the reading today (PpP section 3.2)? What assumption did it start with and how could Dr. Durfee tell that that assumption was wrong?
a. it started by assuming that the wave passed straight from one rope to the next and was wrong because that would lead to the wave having the same velocity on both ropes.
How did the next guess (section 3.3) build on the first?
a. He then guessed that a wave was partially reflected, instead of solely transmitted
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Reflection/transmission at boundaries: The setup
Why are k and the same for I and R? (both labeled k1 and 1) “The Rules” (aka “boundary conditions”)
a. At boundary: f1 = f2
b. At boundary: df1/dx = df2/dx
Region 1: light string Region 2: heavier string
in-going wave transmitted wave
reflected wave
1 1( )i k x tIA e
1 1( )i k x tRA e
2 2( )i k x tTA e
1 1 1 1( ) ( )1
i k x t i k x tI Rf A e A e 2 2( )
2i k x t
Tf A e
Goal: How much of wave is transmitted and reflected? (assume k’s and ’s are known)
x = 0
1 1 1 1 1cos( ) cos( )I I R Rf A k x t A k x t 2 2 2cos( )T Tf A k x t
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Boundaries: The math
1 1 1 1 2 2( 0 ) ( 0 ) ( 0 )i k t i k t i k tI R TA e A e A e
2 2( )2
i k x tTf A e
x = 0
1 20 0B.C.1:
x xf f
1 1 2i t i t i tI R TA e A e A e
I R TA A A and 1 2
1 1 1 1( ) ( )1
i k x t i k x tI Rf A e A e
Goal: How much of wave is transmitted and reflected?
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Boundaries: The math
1 1 2( ) ( ) ( )1 1 2
0 0
i k x t i k x t i k x tI R T
x xik A e ik A e ik A e
2( )2
i k x tTf A e
x = 0
1 2
0 0
B.C.2:x x
df df
dx dx
1 1 2i t i t i t
I R Tik A e ik A e ik A e
1 1 2I R Tk A k A k A
1 1( ) ( )1
i k x t i k x tI Rf A e A e
Goal: How much of wave is transmitted and reflected?
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Boundaries: The math
Like: and
How do you solve?
x = 0
1 1 2I R Tk A k A k A I R TA A A
Goal: How much of wave is transmitted and reflected?
x y z 3 3 5x y z
2 equations, 3 unknowns??
Can’t get x, y, or z, but can get ratios!y = -0.25 x z = 0.75 x
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Boundaries: The results
Recall v = /k, and is the same for region 1 and region 2. So k ~ 1/v
Can write results like this:
x = 0
1 2
1 2
R
I
A k kr
k kA
Goal: How much of wave is transmitted and reflected?
1
1 2
2T
I
A kt
k kA
2 1
1 2
R
I
A v vr
v vA
2
1 2
2T
I
A vt
v vA
“reflection coefficient” “transmission coefficient”
The results….
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Special Cases
Do we ever have a phase shift in reflected or transmitted waves?
a. If so, when? And what is it? What if v2 = 0?
a. When would that occur? What if v2 = v1?
a. When would that occur?
x = 0
2 1
1 2
R
I
A v vr
v vA
2
1 2
2T
I
A vt
v vA
The results….
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Reflected & Transmitted Power
Recall: (A = amplitude)
Region 1: and v are same… so P ~ A2
Region 2: and v are different… more complicated…but energy is conserved, so easy way is:
x = 0
2 21
2P A v
2R
I
PR r
P
21T
I
PT r
P
r,t = ratio of amplitudesR,T = ratio of power/energy
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Clicker question: 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.
2 1
1 2
R
I
A v vr
v vA
2
1 2
2T
I
A vt
v vA
2R r 1T R
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Demo Reflection at a boundary. Measure v1 and
v2.
2 1
1 2
v vrv v
2
1 2
2vtv v
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Now, on to sound!
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Clicker question: Sound waves are typically fastest in:
a. solidsb. liquidsc. gases
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Sound Waves What type of wave? What is waving? Demo: Sound in a vacuum Demo: tuning fork Demo: Singing rod Sinusoidal?
a. Demo: musical disk
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Speed of sound Speed of sound…
a. in gases: ~300-1200 m/sb. in liquids: ~1000-1900 m/sc. in solids: ~2000-6000 m/s
v = sqrt(B/) compare to v = sqrt(T/)
Speed of sound in aira. 343 m/s for air at 20Cb. Dependence on temperature (eqn in
book and also given on exam)
ms343293KsoundT
v
ms343293KsoundT
v