announcements 10/8/10 prayer exam: last day = tomorrow! a. a.correction to syllabus: on saturdays,...
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Announcements 10/8/10 Prayer Exam: last day = tomorrow!
a. Correction to syllabus: on Saturdays, the Testing Center gives out last exam at 3 pm, closes at 4 pm.
Homework problem changes: some extra credit, one moved. See email.
Lab 3 starting tomorrow: it’s a computer simulation. See class website.
a. If we don’t finish the relevant discussion today (which seems likely), you probably should wait until after Monday’s class before starting the lab.
Quick writing assignment while you wait: Ralph is still not quite grasping this… he asks you, “How are complex exponentials related to waves on a string?” What should you tell him to help him understand? (Please actually write down your answer.)
Thought 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: Usually use #3, Aei(-kx-t)
b. Reasons: (1) All terms will have same e-it factor. (2) The sign of the number multiplying x then indicates the direction the wave is traveling.
ˆk k i
Reading Quiz Which of the following was not a major
topic of the reading assignment?a. Dispersionb. Fourier transformsc. Reflectiond. Transmission
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
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?
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?
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
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….
Special Cases
Do we ever have a phase shift? 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….
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
Dispersion A dispersive medium: velocity is different for
different frequenciesa. Any real-world examples?
Why do we care? a. Real waves are often not shaped like sine
waves.– Non sine-wave shapes are made up of
combinations of sine waves at different frequencies.
b. Real waves are not infinite in space or in time.– Finite waves are also made up of combinations
of sine waves at different frequencies.Focus on (b) for now… (a) is the main topic of the “Fourier” lectures of next week.
Wave packets Adding cosines together with Mathematica, “sum of
cosines.nb”
http://www.physics.byu.edu/faculty/colton/courses/phy123-fall10/lectures/lecture%2017%20-%20sum%20of%20cosines.nb
What did we learn?a. To localize a wave in space, you need lots of frequenciesb. To remove neighboring localized waves, you need those
frequencies to spaced close to each other. (infinitely close, really)
Back to Dispersion 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 =4? What is v for =10?
What does that wave look like as time progresses? (Mathematica “dispersion of two cosines.nb”; on website. And next slide.)
Mathematica
0.7 seconds 1.3 seconds
0.1 seconds
Femtosecond Laser PulseFemtosecond Laser Pulse
Et=0=sin(10 x)*exp(-x^2) Power Spectrum
Credit: Dr. Durfee
Initial shape of waveHow much energy is contained in each frequency component ( = vk)
Note: frequencies are infinitely close together
Propagation Of Light PulsePropagation Of Light Pulse
E(x,t) Power Spectrum
Credit: Dr. Durfee
Wave moving in time How much energy is contained in each frequency component
Tracking a Moving PulseTracking a Moving Pulse
E(x+vt,t) Power Spectrum
Credit: Dr. Durfee
Graph “window” is moving along with speed v
How much energy is contained in each frequency component
Laser Pulse in Laser Pulse in DispersiveDispersive Medium Medium
Et=0 = sin(10 x)*exp(-x^2) Power Spectrum
Credit: Dr. Durfee
How much energy is contained in each frequency component
Initial shape of wave
Not all frequency components travel at same speed
Time Evolution of Time Evolution of DispersiveDispersive Pulse PulseCredit: Dr. Durfee
Wave moving in time
Peak moves at about 13 m/s (on my office computer)
How much energy is contained in each frequency component
Tracking a Tracking a DispersiveDispersive Pulse Pulse
E(x+vgt,t)
Credit: Dr. Durfee
Graph window moving along with peak, ~13 m/s